José Orozco-Santiago

Carlos-Armando Cuevas
email: ccuevas@cinvestav.mx

Luc Trouche
email: luc.trouche@ens-lyon.fr

Students' perceptions in a situation regarding eigenvalues and eigenvectors

Keywords: Linear algebra, Eigenvalue, Eigenvector, Scheme, Instrumental genesis

In this study we investigate conceptualization processes among university students in solving a problem involving concepts of eigenvalues and eigenvectors in a linear algebra course for engineering. In the pilot phase of the study, we examined a case study of two first-year linear algebra students and analyzed the data through the theory of instrumental genesis in a dynamic geometry paper-and-pencil environment.

Introduction

Linear algebra is one of the first abstract mathematics courses engineering students take in their early years in college. According to [START_REF] Robert | Quelques resultats sur l'apprentissage de l'algebre lineaire en premiere année de DEUG[END_REF], when first encountering linear algebra, students express concern about the use of formalism, the overwhelming number of new definitions and the lack of connection between this new field and their existing knowledge of mathematics. [START_REF] Dorier | Research into the Teaching and Learning of Linear Algebra[END_REF] distinguish two inseparable sources of difficulty for students in learning processes: the nature of linear algebra itself (conceptual difficulties) and the kind of thinking necessary for the understanding of linear algebra (cognitive difficulties).

After the Linear Algebra Curriculum Study Group (LACSG) recommended the use of technology in early linear algebra courses [START_REF] Carlson | The Linear Algebra Curriculum Study Group Recommendations for the First Course in Linear Algebra[END_REF], some teachers began to incorporate Computer Algebra Systems (CAS) into their classrooms. At the same time, several researchers studied the reflexive use of technological tools in classroom teaching/learning processes (involving both the tool and the mathematical content), which gave rise to the instrumental approach to didactics as a central theoretical framework [START_REF] Artigue | Learning Mathematics in a CAS Environment: The Genesis of a Reflection about Instrumentation and the Dialectics between Technical and Conceptual Work[END_REF]Guin, Ruthven, & Trouche, 2005;[START_REF] Trgalová | Technology and resources in mathematics education[END_REF].

Eigenvalues and eigenvectors are important concepts in the study of linear algebra [START_REF] Gol Tabaghi | How Dragging Changes Students' Awareness: Developing Meanings for Eigenvector and Eigenvalue[END_REF][START_REF] Meel | Tool Building: Web-based Linear Algebra Modules[END_REF], and they require an understanding of other concepts such as vector spaces, linear transformation, bases and dimension, among others. In this paper, we present the preliminary results of our doctoral research, whose objective is to orchestrate a teaching situation appropriate for introducing the concepts of eigenvalues and eigenvectors in a traditional course on linear algebra. For this part of our research, we pose the following question: what utilization schemes does the student experience / develop when faced with a geometry task involving the concepts of eigenvalues and eigenvectors?

Theoretical framework

The French mathematics education community [START_REF] Artigue | Learning Mathematics in a CAS Environment: The Genesis of a Reflection about Instrumentation and the Dialectics between Technical and Conceptual Work[END_REF]Guin et al., 2005) has extended the instrumental approach [START_REF] Rabardel | People and technology: a cognitive approach to contemporary instruments[END_REF] to the learning of mathematics through the reflexive use of technological tools. Rabardel thus defines an instrument as a "mixed entity made up of an artifact and a scheme" (p. 37), retaining Vergnaud's notion of scheme, as "the invariant organization of behavior for a certain class of situations" (Vergnaud, 1998, p. 167). To understand the function and dynamic of a scheme, we must take into account its components. These include one or several goals, each with its sub-goals and anticipations; rules to generate action, information seeking and control; its operational invariants (concepts-in-action and theorems-in-action); and the possibilities of inferences within the situation. Artifacts are associated with two kinds of utilization schemes: usage schemes and instrument-mediated action schemes. Two different users can approach the same artifact differently, develop different utilization schemes, and create two different activities and instruments (Alqahtani & Powell, 2017, p. 14). For a subject, the artifact becomes an instrument through a process, called instrumental genesis, "involving the construction of personal schemes or, more generally, the appropriation of social pre-existing schemes" (Artigue, 2002, p. 250). According to [START_REF] Trouche | Managing the Complexity of Human/Machine Interactions in Computerized Learning Environments: Guiding Students' Command Process through Instrumental Orchestrations[END_REF], instrumental genesis is a complex process that requires time, and is related to artifact's characteristics and the subject's activity. In instrumental genesis, two processes coexist: instrumentalization and instrumentation. "The instrumentation process is the tracer of the artifact on the subject's activity, while the instrumentalization process is the tracer of the subjects' activity on the artifact" (Trouche, 2014, p. 311). In a technological environment, interactions between student and artifact must be organized with specific didactic intentions. The crucial role of teachers in this process was soon recognized. To characterize the teacher's role in guiding students' mastery of tools and their learning processes, [START_REF] Trouche | Managing the Complexity of Human/Machine Interactions in Computerized Learning Environments: Guiding Students' Command Process through Instrumental Orchestrations[END_REF] introduced the notion of instrumental orchestration. An instrumental orchestration consists of two main elements: didactical configuration and exploitation modes.

Methodology

In this paper we present the results of a single task, part of a larger study whose objective is to design an orchestration of a situation to introduce the concepts of eigenvalues and eigenvectors in a first course on linear algebra for engineering students. The task was designed by the teacher, in conjunction with the researcher. The study was carried out in the spring of 2018 and included a case study. The research involved eight volunteer students averaging nineteen years old, all enrolled in a "Fondamentaux des Mathématiques II" class at a public university in France. The session lasted 120 minutes. The students had experience in matrix calculation, vector spaces, and linear transformations, but they had not yet addressed eigenvalues and eigenvectors. They also mentioned having some previous experience with the Dynamic Geometry Environment (DGE), in this case GeoGebra. The students were told that we would be studying the way they solved the problem, or the techniques used, and that they would not be graded. Each student was given the paper assignment and each group (of 2 or 3) was provided with a laptop with GeoGebra and Internet access. The data was collected from observation and field notes by a researcher external to the experiment, and lasted about 120 minutes. For our analysis, we selected two students: Cécile and Henry.

Mathematical context and task design.

An eigenvector of an matrix is a nonzero vector such that for some scalar . A scalar is called an eigenvalue of A if there is a nontrivial solution of ; such an is called an eigenvector corresponding to [START_REF] Lay | Linear algebra and its applications[END_REF]. Here we consider , by the availability of a DGE (GeoGebra), considering the potentiality of the dragging tool "wandering dragging" [START_REF] Arzarello | A cognitive analysis of dragging practises in Cabri environments[END_REF].

The task (see figure 1) was not designed to evaluate the student's progress in the course. Rather, it was restricted to having the students to work on in order to study their conceptualization of eigenvalues (real) and eigenvectors of 2x2 matrices, specifically in solving the first two points in a single class period of 30-50 minutes.

Given the following two rectangles.

1. Find a process to build a linear application that sends the rectangle OABC to the rectangle ODEF. 2. Apply this process when the points have the following canonical coordinates:

And build the matrix of in this basis 3. In GeoGebra, we code a matrix with the braces . We can create a then move its end and apply the matrix by . Model the previous situation and check the values. 4. Can we find a vector such that for a certain real ? What are the possible values of ? What are the possible vectors for a given ? 5. Let and . Give the matrix of the linear application in this database.

Figure 1. The task discussed in this document

The students are invited to develop intuitive notions of eigenvalues and eigenvectors associated with the matrix M, in the environment of GeoGebra, by means of a draggable vector and its vector image , as the vector is dragged around the screen, the vector moves accordingly. To find an eigenvalue of the matrix M geometrically, the student will have to drag a position where and are collinear, in order to explore the relationship involved in the equation , and to recognize that there is an infinite number of eigenvectors associated with each eigenvalue.

Instrumental orchestration of the class

The main objective of the orchestration is to introduce and use the concepts of values and eigenvectors in a first course on linear algebra. The secondary objective is to analyze a geometric problem in a paper-pencil environment and in a DGE to explore the problem and validate the solutions proposed by the students. It is defined by a didactical configuration (students organized in heterogeneous working groups, with access to GeoGebra) and exploitation modes (each group analyzes the problem and proposes solutions using a combination of paper-and-pencil work and GeoGebra; the group is free to ask questions of the teacher and each participant's role is identified).

Data Analysis

The students followed the steps given in the task. Question # 1 can be identified as the phase of data selection, solution strategies and operation to be performed to solve the task:  A transformation (matrix) that stretches the rectangle horizontally and another transformation (matrix) that stretches it vertically, then a composition of transformations;  Display a matrix of that and (matrix-vector);  Display a matrix of that and (matrix-vector-scale),

 Discover the matrix by trial and error with the help of GeoGebra (trial-and-error).

First Case: Cécile.

The components of Cécile's "matrix-vector" scheme were:

Goal: find a linear transformation, such that T(OABC)=ODEF Sub goals:

-Identify the possible relations of parallelism and collinearity -Recognize the proportionality constant k , such that and -Check that the result fulfills the conditions of the problem.

Anticipations

Know that you have more than one way to solve the problem--either DGE or paperand-pencil.

Rules of action -If I find a matrix that when multiplied by , gives , then that same matrix must when multiplied by , give . -If I find a matrix and enter it into GeoGebra, I can test the solution.

Operational invariants

Concepts-in-action

-Two vectors are collinear if they can be placed one on top of the other or they are on the same straight line Theorems-in-action -TA1 If when moving the vector both clockwise and counterclockwise, the opening (angle) between the vector and the vector decreases so that the vector superimposes the vector , then the matrix M has an eigenvector -TA2 If the opening (angle) between the vector and the vector does not decrease when the vector moves both clockwise and counterclockwise, then the matrix M does not have an eigenvector.

Possibilities of inference

-If I find a matrix M that when multiplied by gives , there is a 3/2 relation.

Table 1: Elements of Cécile's utilization scheme

Cécile began by using the paper-and-pencil technique. From the rectangle OABC, she identified and as the vectors: and . In the rectangle ODEF, she also identified y as the vectors: y . She defined a 2x2 matrix which multiplied by the vector must give the vector , and which multiplied by must give the vector (see figure 2). From the two systems of linear equations obtained, Cécile grouped the equations with the same unknowns into two 2x2 systems and, after solving the two systems, built the matrix . Cécile said she had rarely used GeoGebra in her previous courses, so she requested support from a partner to be able to input the matrix. Afterwards, she input the operations M*C, M*B and M*A, creating points G, H and I (names automatically assigned by GeoGebra). When entering the operations, she observed that the points obtained did not coincide with the points D, E and F. In fact, she observed that two of the three points did not coincide. She first checked her notes to see if she had made some arithmetic error but was still unable identify the problem, so she turned to the teacher for support. The teacher reviewed Cécile's worksheet and pointed out an arithmeticalgebraic error. Cécile then corrected the solutions to the equation system, obtaining the matrix . Cécile entered this matrix in GeoGebra and performed the operations D'= M*A; E'=M*B and F'=M*C. This time the points coincided, so the matrix M obtained was considered acceptable.

Proceeding on to question 3, Cécile focused her attention on the edges of the rectangles, mentioning that "while I drag the vector towards the base of the rectangle, the vector follows the vector and reaches it at the base" and "while I drag the vector towards the height of the rectangle, the vector pursues the vector and reaches it exactly at the top." She continued, "when the vector is exactly the same as the base OC, the vector is twice , i.e. if then , but when is approximately the base OC, there is a difference of hundredths, between and twice , i.e. if then (indicated in the Algebraic View). Cécile is able to observe that when , and that only then is ; with other values, she notes, fails to reach twice , due to decimals. Cécile fails to calculate the other eigenvalue and fails to intuit the concepts of eigenvalue and eigenvector.

At this point, we observe that in Cécile's instrumentation process, the tool guides her attention and the instrumentalization process is recognized by the "wandering dragging" functionality. Gol Tabaghi called this dragging modality "intentional dragging--which involves dragging a point with the intention of producing a certain configuration" (p. 234).

Second Case: Henry.

The components of Henry's scheme are:

Goal: find a linear transformation, such that T(A)=D, T(B)=E, T(C)=F.

Sub goals:

-Identify the possible relations of parallelism and collinearity -Verify that the result fulfills the conditions of the problem.

Anticipations

Know that you have more than one way to solve the problem--either DGE or paperand-pencil.

Rules of action

-If you succeed in matching point E' with point E, you will have found the matrix M.

Operational invariants

Concepts-in-action -Two vectors are collinear if they can be placed one on top of the other or they are on the same straight line Theorems-in-action -If the vectors and are linearly independent, then the matrix M does not have an eigenvector -If the vectors and re linearly dependent, then the matrix M has an eigenvector.

Possibilities of inference

-If the vectors and are dependent, then .

Table 2: Elements of Henry's utilization scheme

Henry used a trial-and-error scheme with DGE. Since Henry had experience with GeoGebra, he created four sliders--x1, x2, x3 and x4--and then, with the sliders, he created a 2x2 matrix M:

. The interval that he defined for the four sliders was -5 to 5, increments of 0.1; finally he introduced the three points E', D' and F' (see figure 3).

Henry commented that if he could match point E' with point E, the other points (D' and F') would be superimposed as well. On his first attempt, Henry managed to match point E' with point E (see figure 3), but contrary to what the thought, the other two points (D' and F') did not overlap. Subsequently, through trial and error, by moving the sliders, he managed to obtain the matrix M. Henry mentions that the elements of the matrix are decimals, which could cause the points to "overlap, but they are not equal." At this point we believe that Henry saw points E and E' overlapping, and noted that their values were the same in Algebraic View , but the E and E' do not overlap exactly, i.e., they show a slight offset. Henry then turned to the GeoGebra Relation1 tool to verify that they were in fact the same. He did not use paper-and-pencil.

To resolve question 3, since Henry had already entered the matrix M in GeoGebra, he entered only the vector and . Then he began to drag clockwise. Henry used the "Algebraic View" in GeoGebra to observe the numerical values and relied on the Relation tool to explore the relationships between and . As the vector was brought closer to the segment, he used the Zoom tool to make sure that they were able to overlap, and continued until the two vectors and lay one on top of the other. Henry said: "of course, and are collinear," and to avoid further approximation he modified the vector . "When is (2,1)," Henry continued, "the vector is twice ." He then wrote , stating that 2 is . With this same analysis he once again modified the value of and said: "uff, they're decimals, I think the value is....I'd better check it..." Henry then used the Distance or Length tool to calculate the norm of and . And after calculating the rules he said that in fact . Once finished, Henry began working with the "matrix-vector-scalar" scheme using paper-andpencil. In obtaining the linear equation systems, Henry represented the systems as lines in the plane and obtained the values through their intersections, concluding satisfactorily with the matrix M. We conclude that Henry displays a good level of instrumentation and instrumentalization of the DGE.

Preliminary conclusion

Using the above tasks, we were able to observe the schemes that students develop, and the difficulties they may encounter, when using the concepts of eigenvalue and eigenvector before formally addressing them in class and when the problems arise outside the context in which they have been taught (given a matrix, obtain its eigenvalues and eigenvectors). The conceptualization of eigenvalues and eigenvectors is a long and continuous process, in which the teacher plays an important role by carefully choosing situations that allow the mathematical knowledge to be meaningful. We believe that the success of Henry's scheme was possible because the elements of the matrix M are exact, and they lay within the interval that he had defined from the beginning (-5 to 5). When dragging and observing the change in , the participants were prompted to look for relationships between these two objects and relate them to the matrix obtained. The use of GeoGebra helped Henry with his conceptual difficulties, while Cécile showed a better conceptual knowledge, relying on geometrical thinking. In this experience we were unable to determine whether the students recognized the existence of an infinity of eigenvectors associated with each eigenvalue. Our future research will study two questions: What orchestrations can help the teacher foster the students' development of conceptual knowledge? And, how can students be helped to recognize how many different eigenvectors are associated with a given eigenvalue?

Figure 2 .

 2 Figure 2. Cécile's working to find the matrix M

Figure 3 .

 3 Figure 3. Henry's technique for finding the matrix M

This command shows, in a message box, the relationship between two objects. It allows you to verify whether two lines are perpendicular/parallel, two (or more) objects are equal, or three points are collinear, among others.(https://wiki.geogebra.org/en/Relation_Command)

Acknowledgment

We are grateful to Prof. Christian Mercat and the students of the University Claude Bernard Lyon 1, for allowing us to conduct this research.


Xavier Nicolas
email: xavier.nicolas@uca.fr

Jana Trgalova
email: jana.trgalova@univ-lyon1.fr

A virtual environment dedicated to spatial geometry to help students to see better in space

Keywords: Mathematics, learning, geometry, visualization, technology

We report about an experiment carried out in a school context of teaching spatial geometry with a dynamic geometry environment, based on gesture-based interface for use with immersive, roomscale virtual reality. The work reported in this paper is the first iteration of a research cycle involving teachers, researchers and developers to help high school students better "see" in space and propose principles for the design and use of such environments in a school context. Our study focuses on the impact of "dimensional construction" techniques on the way students perceive a geometric figure represented in a cavalier perspective. Our first results show a significant improvement of students' performance, demonstrating the appropriation and transfer of these dimensional construction techniques to 2D supports as well as the transition from an iconic to a non-iconic vision of figures, which is an essential condition for geometric activity.

Introduction

Teaching geometry in space is considered difficult because it is necessary to "see in space," objects or situations most often represented by drawings in cavalier perspective. In geometry, dynamic geometry environments (DGE) have revolutionized not only the way geometric objects are created and manipulated, but also allowed new didactic situations and modified our very relationship to geometric objects [START_REF] Laborde | Technology used as a tool for mediating knowledge in the teaching of mathematics: the case of Cabri-geometry[END_REF]. However, DGE are constrained by a 2D representation mode (screen), which poses perception and usability problems [START_REF] Dimmel | Handwaver : a gesture-based virtual mathematical making environment[END_REF]. In the domain of technology in education, unlike the professional world, virtual reality (VR) has made a timid entry. A DGE in VR could be relevant to better see in space or at least avoid the problem, but what will happen once the HMD (Head Mounted Display) is removed and the students are back on their tables? An interesting aspect of virtual worlds that can be exploited is their modes of interaction and assistance that are impossible in the physical world as a way of approaching learning differently [START_REF] Burkhardt | La réalité virtuelle pour l'apprentissage humain[END_REF]. VR can revel and make tangible concepts or information normally outside the field of human perception [START_REF] Winn | Beyond Constructivism: A Return to Science-Based Research and Practice in Educational Technology[END_REF]. This leads us to question whether concrete and intelligible manipulation of geometric objects performed by students in a VR environment can change the way they later see them in cavalier perspective drawings? In this paper, we present a research aiming to evaluate the potential of VR environment in geometry in space, but also to set some benchmarks for academic learning.

Theoretical framework

Seeing in space in geometry

In the context of geometry, the act of seeing involves complex perceptive and cognitive processes. For [START_REF] Duval | Les conditions cognitives de l'apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements[END_REF], a major cognitive problem is how to move from discriminative recognition of shapes to the identification of objects to be seen. This transition can take two paths: an iconic and a non-iconic path. The iconic path is based on the similarity of the shapes with the object and is not specific to geometry. Seeing in space in an iconic way can also be considered from the perspective of spatial skills as "the ability to generate, preserve, recover and transform well-structured visual images" (Lohman, 1996, p. 98). The non-iconic way relies on considering a representation of figural units linked by relations as a whole. Most of spatial geometry tasks imply this flexibility of vision. The dimensional deconstruction [START_REF] Duval | Les conditions cognitives de l'apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements[END_REF], i.e., considering the figural units of smaller dimensions in a geometric object, is the central point for the geometric processing of representations. Following Duval, we start from the premise that getting students to mobilize dimensional deconstruction on a 3D object represented in 2Dcan help them better "see in space".

According to [START_REF] Chaachoua | Fonctions du dessin dans l'enseignement de la géométrie dans l'espace[END_REF], to be efficient in geometry, the representations must fulfill three functions: illustration, supporting assumptions and allowing the heuristic change of point of view like the dimensional deconstruction. In spatial geometry, the drawings in cavalier perspective are the most commonly used representations. Mithalal (2014) notes that drawings in perspective struggle to fulfill these functions, and that dynamic geometry environments (DGE) allow them to be partially restored.

In order to tackle our initial question, we need to identify different ways of seeing in space students mobilize in a geometric activity. This involves modelling the ability to "see in space" in an ecological approach, which leads us to choose the anthropological theory of didactics.

The anthropological theory of didactics

The anthropological theory of didactics (ATD, Chevallard, 2012) provides an epistemological framework aiming at the understanding of the ecology of mathematical knowledge. In ATD, any human activity is modelled through the notion of praxeology represented by so called "4T-models (T,τ,θ,Θ)". A type of task (T) and the relevant techniques (τ) for solving tasks t of the type T constitute a practical block or praxis (know-how). The technologico-theoretical block or logos (know why) covers a technology (θ) explaining and justifying the technique and a theory (Θ) justifying the underlying technology. In the case where the type of tasks T relates to the field of mathematics, we use the term of mathematical organization (MO) rather than praxeology. For Duval, "the way of seeing a figure depends on the activity in which it is mobilized" (Duval, 2005, p. 8). From the ATD perspective, we consider the different ways of seeing as intrinsic elementary praxis (EP) because, on the one hand, they only exist through their mobilization within techniques of prescriptible types of tasks and, on the other hand, they are present at a level of granularity that the didactics of mathematics can no longer exceed. We propose the following elementary praxis to describe the possible transitions from a figure to a geometric object. EP1 and EP2 follow the iconic way [START_REF] Chaachoua | Fonctions du dessin dans l'enseignement de la géométrie dans l'espace[END_REF], EP3 the non-iconic way [START_REF] Duval | Les conditions cognitives de l'apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements[END_REF].

 EP1: Associate the recognized shapes with the appearance of a physical object seen from a certain point of view (identification by natural observation and resemblance).  EP2: Associate the recognized shapes with a prototypical image of the object available in a mental catalogue (identification by prototypical shapes).

 EP3: Identify figural units and their relationships (identification by processes supported by dimensional deconstruction).

The relationship to knowledge

In ATD, the notion of relationship to knowledge [START_REF] Chevallard | Concepts fondamentaux de la didactique : perspectives apportées par une approche anthropologique[END_REF] leads to considering the didactics from the anthropological perspective. An individual's relationship to particular (piece of) knowledge may be different from the relationship that the educational institution has with that knowledge. In the case of the ability to see in space, the student develops his/her personal relationship within and outside the school walls. The analysis of student productions, and in particular of their mistakes with regard to the different EPs mobilized in the techniques used to solve a geometric task, can provide us with information on the student's personal relationship with the notion of seeing in space. At the secondary level, in geometry, as well as more broadly in mathematics, the curriculum clearly aims to develop a deductive approach. From the point of view of the institutional relationship to "seeing in space", EP1 and EP2 are non-compliant and errorprone relationships. It is the EP3 that is targeted by the teaching of geometry in space.

The concept of ostensive/non-ostensive

The implementation of a technique involves manipulation of ostensives, i.e., tangible and manipulable objects, regulated by non-ostensives, i.e., theoretical objects as concepts, notions, ideas [START_REF] Bosch | La sensibilité de l'activité mathématique aux ostensifs[END_REF]. Ostensives can belong to different registers: graphic, gestural, discursive and scriptural. The performance of an ostensive can be evaluated according to two criteria: its instrumental valence, which is its ability to act, to work and its semiotic valence, which is its ability to see, to appreciate in a sensitive way the work done or to plan the work to be done.

According to Chevallard (1994, p 8), "one of the factors of progress in mathematics is the creation of ostensives that are effective from both an instrumental and a semiotic point of view." (our translation).

We have seen in the introduction that drawings in perspective are ostensives whose instrumental and semiotic valences are not sufficient to develop a non-iconic way of seeing in space.

Analysis of Handwaver, a DGE in virtual reality

The Handwaver application is developed by a research team at the University of Maine to exploit the modes of representation and interaction available in virtual environments to create experiences where learners use pseudo natural gestures to observe, create and manipulate mathematical objects [START_REF] Dimmel | Handwaver : a gesture-based virtual mathematical making environment[END_REF]. In Handwaver, the "stretch" and the "extrude" operators allow to generate geometric objects of n dimensions from an object of n-1 dimension (Figure 1). As we focus on 3D objects, we consider the generation of a polyhedron from a polygon as a type of task T, made up of several punctual mathematical organizations (PMO), i.e., relating to a single subtask of T. We describe these praxeologies and the involved ostensives in Table 1.

Theory

Euclidean geometry Euclidean geometry

Ostensives

The stretch operator The extrude operator

Graphical register

Virtual model of the prism reacting in real time to the manipulation of polygonal bases

Virtual model of the pyramid reacting in real time to the manipulation of the apex and the polygonal base

Gestural register

Grasp and stretch (with hands apart) the polygon Touch the polygon with the extrude tool and move away the apex created (with hands apart)

Discursive register

(later introduced by the teachers of our experiment) "I stretch the polygon" "I pinch the polygon"

Table 1: Two examples of Punctual Mathematical Organizations related to the type of task T

The a priori analysis of the stretch and extrude ostensives leads us to make the following observations:

 From the point of view of their instrumental valence, the mobilization of pseudo natural gestures makes the techniques easy to apply and retain.  From the point of view of their semiotic valence, the environment provides immediate feedback on the manipulation of objects in the form of visualization of the polyhedron being generated. This enables the student to control the process of generating polyhedrons, and thus makes the techniques, and more generally the continuity of the process leading from the polygon to the polyhedron, readable and intelligible.

With regard to these two considerations, we hypothesize that these techniques have a strong reversibility potential, i.e., they allow a dimensional deconstruction of the polyhedrons enabling to identify the starting polygon. This leads us to reformulate our initial questioning:

To what extent the use of Handwaver to solve tasks involving dimensional construction techniques help students to move from an iconic (EP1 & EP2) to a non-iconic (EP3) way of seeing drawings in a cavalier perspective?

Methodology

Our methodology relies on a teaching experiment involving two grade 6 classes and their mathematics teachers. Our aim was to design and test activities with HandWaver involving the above mentioned techniques. The methodology comprises several phases outlined below.

Phase 1: Identification of errors related with seeing in space

We interviewed six secondary school mathematics teachers through two focus groups in two schools to determine types of tasks where seeing in space is problematic for students. We present here two errors that are widespread according to the teachers at the beginning of secondary school and that persist throughout compulsory schooling:

 Error 1 (E1): The student does not consider cubes or rectangular cuboïds as prisms.

 Error 2 (E2): The student does not identify properly the base of a prism or a pyramid in a non-prototypical position.

From the point of view of our theoretical framework, E1 and E2 are characteristic of a relationship of knowledge that does not conform to the one expected by the institution. EP1 leads for example to difficulties with perceiving geometric properties shared by different solids as this requires identifying components or figurative units and their relationships, which EP1 does not allow. EP2 comes from observation of real objects and leads for example to always identify the base of a solid as a face on which the solid sits on. E1 and E2 errors prove the non-mobilization of EP3.

Phase 2: Diagnostic assessment

To investigate these statements, we analyzed the outcomes of a diagnostic assessment administered to 40 grade 6 students (11-12 years old, two classes) who have not had yet teaching of geometry in space. The test consisted of two exercises: classify given solids by family and identify a base for each of the given solids. The solids are represented by drawings in cavalier perspective. E1 and E2 errors were present in all students, confirming the statements collected from the focus group teachers. We note that only EP1 and EP2 are present in students' personal praxeology for these types of tasks. We assume that we can evaluate changes in ways of seeing in space with respect to the presence of these errors in students' productions. We can therefore test the following hypothesis:

The dimensional construction technique makes it possible to develop the EP3 praxis in the students' personal praxeology.

Phase 3: Design and implementation of tasks in Handwaver environment

With the two teachers, we have designed a scenario to allow students to implement the dimensional construction PMOdim_prism and PMOdim_pyramid (Table 1). The scenario outlined below was implemented by each of the two teachers in their 6th grade class of 20 students who passed the diagnostic assessment and whose results are reported above. It should be noted that this was the first session where the notions of prisms and pyramids were introduced at this level of class. In terms of MO, the type of task underpinning the first exercise is "Identify prisms and pyramids in a set of solids in cavalier perspective"(T1) and the type of task underpinning the second one is "Identify the base of solids in cavalier perspective" (T2). The test includes 12 solids represented in cavalier perspective whose visible faces are colored to avoid perceptual aberrations. Among the solids are 5 prisms including a cube and a rectangular cuboid, 4 pyramids, 1 cone, 1 cylinder of revolution, and 1 sphere (Figure 3). A particular attention was paid to draw some of the solids in a non-prototypical position.

Findings and discussion

In the analysis of the post-test results, we focus on those that are relevant to the types of tasks T1 and T2. We obtain the following success rates:

 79% of students correctly solved T1 (68% for prism and 93% for pyramids).  75% of students correctly solved T2

If we consider only the object in the post test who can generate the errors E1 and E2:

 Error 1 (E1) 70% of students correctly solved T1 (Fig: 1, 2)  Error 2 (E2): 59% of student correctly solved T2 (Fig: 3,4,7,9,10,12).

Even if we have not submitted a real pre-test, only a diagnostic assessment, the very significant decrease in the presence of E1 and E2 in student production seems indicate the non-mobilization of EP1 or EP2. The new technique used by student is more relevant, that could indicate the mobilization of EP3 and could be link to the PMOs used in the VR environment. This seems to confirm our hypothesis and the a priori analysis of stretch and extrude ostensives in terms of their instrumental and semiotic valence. However the solids generation by extrusion is not new, it is often used in an imaginative way by teachers or in CAD software used by students in technology teaching. Yet mistakes persists. The novelty in VR lies in the use of a behavioral interface that affects the graphical and gestural registers of the ostensive. We must consider how cognition operates within constraints imposed by our physiology. First about the graphical register, artificial environments can use computer technology to create metaphorical representations in order to bring to students concepts and principles that normally lie outside the reach of direct experience. [START_REF] Winn | Beyond Constructivism: A Return to Science-Based Research and Practice in Educational Technology[END_REF]. Secondly about the gestural register, in the embodied cognition approach, the mathematical concept of dimensional construction that we mobilized in our experiment, could be considered as a conceptual metaphor (Nunez, 2009), grounded in bodily-based mechanisms : the stretch gesture like stretching gum from a ball to a thread (from a 0D to 1D object), or pulling a blind down (from a 1D to 2D object). So we can suppose in our case that students can easily accept that stretching a polygon makes a 3D object so they can mobilize spontaneously this technique later.

Conclusion

The VR activity implemented seems to have helped students develop a heuristic understanding of drawings in cavalier perspective, based on the identification of figurative units of smaller dimensions, and thus helping to better see in space. In a VR environment we call presence the belief that you are "in" the artificial environment, not in the laboratory or classroom interacting with a computer. The coupling effect leads to a permeability at the cognitive level between the real world and the artificial world. An environment considered as real by users but where we program the interactions and rules, open a field of great opportunities for learning.

Figure 1 .

 1 Figure 1. Different cases of the "stretch" operator: a point is stretched into a line segment, the segment is stretched into a polygon, and the polygon is stretched into a prism (Dimmel & Bock, 2017)

  of a polygon and join corresponding vertices of the two polygons by segments generating faces of the prism Extract an apex from the polygon, pull it out of the plane and join it with all vertices of the polygon generating faces of the pyramid Technology Definition of a prism: a polyhedron with two polygonal faces lying in parallel planes and with the other faces parallelograms Definition of a pyramid: a polyhedron with a polygonal base and three or more triangular faces that meet at a point, called apex

Figure 2 .

 2 Figure 2. The classroom setting and the generation of a prism using the stretch operator The implementation went through the following phases in the classroom setting shown in Figure 2 (left):  Introduction of the activity and demonstration of the environment by the teacher  Generation of solids in the environment by students with the stretch or the extrude operatorone student manipulated at a time, the others observed the screen (Figure 2, right)  Classification, by each student, of the solids generated on the basis of the techniques used by the group (mental re-mobilization and reversibility of techniques)  Discussion, argumentation, validation of the students' classifications (development of a discourse on techniques) -whole class  Institutionalization by the teacher of prisms and pyramids in relation to the techniques used  Mental reinvestment of the techniques considering a disc (to generate either a cylinder or a cone) -whole-class discussion Phase 4: Post-testA post-test was administered to the students 10 days after the session. As the students were on a school trip during these 10 days, they did not have any mathematics lessons during this period.

Figure 3 .

 3 Figure 3. The set of solids from the post-test The post-test includes the following two exercises:  Classify the solids in the families of prisms, pyramids, or other.


Caroline Bodolec

Su Weilin

TECHNOLOGY AND PATRONAGE OF CONSTRUCTION PROJECTS IN LATE MING CHINA -THE CASE OF THE ITINERANT MONK-ARCHITECT MIAOFENG FUDENG (1540-1613) 1

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

second part is devoted to a larger history of Chinese brick-vault architecture and to the place of Buddhist monks in building activities. Here, I give a background for Miaofeng Fudeng's place in both the social and technological history of construction in China. 6 In the last part, I try to show how his building activities can be seen in an empire-wide context favorable to the expansion of Buddhism. I investigate the way the benevolence toward religion manifested in the architecture and the restoration of temples with the support and patronage of the local gentry, along with high-ranking members of the administration and especially the Empress Dowager Cisheng herself.

1-The path of a monk-architect

1-1 Build to order

The first building attributed to Miaofeng Fudeng is situated on Luya Mountain 蘆芽山 in Shanxi province. It is said that at the end of 1582 the Empress Dowager Cisheng, mother of the Emperor Wanli (1546-1615) ordered him to build the main temple of Luya Mountain, the Huayan monastery 華嚴寺 and to build a seven-story iron pagoda at the top of the mountain, the Wanfo pagoda 萬 佛 鐵 塔 . 7 The association of bronze and brick constructions occurred at least three other times in Miaofeng Fudeng building sites. Some time later, Wang Chonggu 王 崇古(1515-1588) 8 asked Miaofeng Fudeng to leave Luya Mountain and go to the Wangu monastery 萬 固 寺 , a monastery in Puban 蒲坂 about 600 km away to the south. 9 This high official, a native of Puzhou 浦州, in that same area, asked him to restore two sites within the temple. There already existed a thirteen-story Tang dynasty (618-907) pagoda 10 and a building called Dafo dian 大 佛 殿 (Great Buddha hall). It is 6 Construction history differs from the history of architecture in its approach to the buildings. Researchers in construction history try to link the history of styles and forms with the history of construction materials, architectural techniques, and the artisans who worked on building sites. One can consult the website of Construction History Society founded in Cambridge (UK). The journal Construction History exists since 1985. The academic community meets regularly at international conferences since the first conference in Madrid in 2003. The next will be organized in Brussels in 2018. http://www.constructionhistory.co.uk/ [consulted 2018-03-23] 7 These buildings were destroyed during the battles that brought the Ming dynasty to an end. See the Gazetteer of Shanxi province, quoted by Glahn, "Fu Deng," in Goodrich and Fang eds., 1976, pp. 462-466; see esp. p. 463. In Kefu 柯夫 and Qianqing 倩青, 1990, pp. 103-104, it is said that Miaofeng Fudeng asked the empress dowager for help with the project. Weidner, 2001, p. 130, note 46. 8 Wang passed the jinshi 進士 examination, the highest level of official examinations, in 1542-1543. At the time he would have held the position of Minister of War (Bingbu shangshu 兵部 尚書). See Zhang Huizhi 張撝之 et al., 1999, p. 216;and Goodrich and Fang, eds., 1976, pp. 1369-1374. Glahn, 1976, p. 463. 9 West of modern Yongji 永濟, Yuncheng 運城 district, near Shaanxi and Henan provinces 10 The text says that the pagoda measured 300 chi 尺 (96 m). Under the Ming dynasty, 1 chi recorded that Miaofeng Fudeng stayed there for three years working on this restoration. An inscription on the pagoda dates it to 1586. The extant ruins of the building attributed to Miaofeng Fudeng show us what could be his first beamless hall. [START_REF] Zhongguang | The Baohua shan gazetteer records that the height of the pagoda was 13 zhang 丈 (equivalent to 43.16 m); "Miaofeng Chanshi zhuan[END_REF] The building was destroyed during the Tongzhi era (1862-1875) of the Qing dynasty.

When the constructions at the Wangu monastery were completed, Miaofeng Fudeng was asked by the Minister of Justice himself, Li Shida 李世達 (1533-1599), [START_REF]A native of Jingyang in Shaanxi province[END_REF] to supervise the repair of a bridge on the Wei 渭 River in the district of Sanyuan 三原 in Shaanxi province. This bridge apparently had thirteen arches. [START_REF]Miaofeng Chanshi zhuan[END_REF] It seems that Miaofeng Fudeng spent two years on the building site and then decided to return to Luya Mountain On his way back, at a place called Ninghua 寧化 in the territory of Ningwu 寧武 district, Miaofeng Fudeng spotted a vertical cliff. It was the perfect place to construct a cave temple decorated with carved figures of the ten thousand Buddhas and stories from the Huayan jing 華 嚴 經 (Flower Garland Sutra). According to the Ningwu gazetteer, an indication of the dates is given: it took three years to excavate and carve the temple, from 1591 to 1594. [START_REF] Deqing | lower part[END_REF] Between 1594 and 1599, Miaofeng Fudeng received a summons from the Governor of the Northern Metropolitan Province (Bei Zhili 北 直 隸 ), [START_REF]The Northern Metropolitan Region included parts of the actual provinces of Hebei, Henan, Shandong and some districts of Beijing and Tianjin[END_REF] Wang Xiangqian 王象乾 (ca. 1544-1626), [START_REF]A native of Xincheng 新城 in Shandong province[END_REF] to supervise the construction of a bridge over the Yang 洋 River in the Xuanhua district 宣 化 縣 , in northern Shanxi province, near the Mongolian border. This was a strategic place where the river was unpredictable, dangerous and very wide. It was certainly because of his previous success with the Sanyuan Bridge on the Wei River that Wang asked Miaofeng Fudeng to take on this challenge. [START_REF]At least, this was mentioned in the Baohua shan gazetteer[END_REF] It is recorded that the bridge had twenty-three arches; but nothing remains of it today. [START_REF] Deqing | lower part[END_REF]

1-2 Build to fulfill a vow

Hanshan Deqing's text indicates that Miaofeng Fudeng was by then focused on a vow he had made a few years before, to offer three statues and three bronze pavilions (tongdian 銅 殿 ) to three sacred mountains: Mount Wutai 五 臺 山 , Mount Emei 峨 嵋 山 and Mount Putuo 普 陀 山 . In the spring of 1599, 19 Miaofeng Fudeng went to visit Prince Shen 瀋 , 20 who lived at Lu'an 潞 安 (modern-day Lucheng 潞城) in the South of Taiyuan 太原. 21 At that time, the Prince intended to offer a statue to a Mount Emei monastery. Miaofeng Fudeng presented to him the project of the bronze buildings, indicating that each of them would cost 10,000 jin 金. Prince Shen agreed to finance the one for Mount Emei and accompanied Miaofeng to Jingzhou 荆 州 in Huguang 湖 廣 . 22 With the assistance of Prince Shen's staff, the bronze elements were cast and sent to Sichuan province. Miaofeng himself went to Mount Emei.

At that time, Wang Xiangqian, Grand Coordinator (dazhongcheng 大中丞) of Sichuan province, was very pleased with the statue and the bronze pavilion and decided to finance the work on the Yong'an monastery 永 安 寺 . The beamless hall known as the Wannian monastery 萬 年 寺 was part of this project. 23 Wang Xiangqiang also agreed to finance the statue for Mount Putuo, the sacred island off Ningbo 寧波, on the coast of Zhejiang province. Around 1603, Miaofeng returned to Jingzhou to oversee the casting of the statue and the pavilion. Everything was ready, and the bronze pieces were already on their way to the sacred island, when, at a place called Longmen 龍門, 24 Miaofeng Fudeng learned the bad news that the monks of Mount Putuo had refused the gifts. One reason for this refusal was the fear of so-called Japanese pirates were active along the coast. 25 Looking for a good place to erect the statue and the pavilion, at the end of 1603, Miaofeng Fudeng finally arrived at Mount Baohua 寶華山 in the actual Jiangsu province, thirty kilometers from Nanjing. As this setback put a serious strain on the budget, Miaofeng Fudeng presented a special request to the Empress Dowager. He immediately received the sum he requested so that he was able to build not only the bronze structures but also two little pavilions of bricks. 19 Wanli jihai year, spring 萬曆己亥春. Hanshan Deqing,in Xu Zangjing,p. 675,lower part, Zhu Xiaoyong 朱效鏞. 21 District of Changzhi 長治. 22 At this time, Hunan and Hubei provinces were joined; the city is in modern Hubei province. 23 shan tuzhi, 1935, juan 5, p. 13b. Zhang Dewei, 2016, p. 358. 24 Also referred to as Jinling 金陵 in the Emei shan gazetteer; Emei shan tuzhi, juan 5, p. 13a. 25 The biographies only mention the refusal of the Putuo shan monks: Putuo seng liju zhi" 普 陀僧力拒之 [Putuo monks strongly rejected them], Hanshan Deqing, in Xuzang jing, p. 676, upper part, ligne 3-4. The hypothesis of the fear of pirates is proposed by Gu Yue, 1997, p. 25. We have not yet found any mention of this incident in the Putuo Shan gazetteers.

"Lidai gaoseng [Ming] dazhi heshang: Miaofeng" 歷代高僧 [明] 大智和尚妙峰, Emei
It is reported that the buildings of the Huiju monastery 慧居寺, also known as the Longchang monastery 隆昌寺,were finished in 1605. [START_REF]Wenshu wuliang dian, Puxian wuliang dian[END_REF] The money sent by the Empress Dowager was not only intended for Mount Baohua but also for Mount Wutai. In the spring of 1605, [START_REF] Wanli Yisi Year | spring 萬曆乙巳春. 28 Wanli bingwu year[END_REF] Miaofeng Fudeng left for the Xiantong Monastery in Mount Wutai. The Empress Dowager sent two eunuchs to help and to oversee the use of funds. One was Wang Zhong 王忠, from the Directorate of the Imperial Horses Stables yuma jian 御 馬 監 ; the other was Chen Ru 陳儒"eunuch from the imperial entourage" (jinchi dajian 近 持大監). Work began in June 1606; several buildings, all made of brick, were constructed. 28 These include a vast beamless hall that is 28 meters wide, 16 meters deep and 20 meters high, surrounded by two smaller beamless halls and a bronze pavilion (Figs. 1 and2). 29 By imperial decree, the monastery was renamed the Shengguang Yongming monastery 聖光永明寺 . When the buildings were finished, Miaofeng Fudeng organized a great ceremony with ten eminent monks who recited the Huayan jing (Flower Garland Sutra). The costs were covered by public funds and contributions from the local population.

1-3 Build to the end

Miaofeng Fudeng was then in charge of the restoration of bridges on the western road to Mount Wutai via Fuping 阜 平 in the Northern Metropolitan Province. He built rest houses for pilgrims, where they could benefit from tea and medicine distributed at the expense of the emperor. Texts record that all the constructions were again made of brick30 . In the same period, and probably linked with the road and bridge projects, the Empress Dowager ordered Miaofeng to restore the Ciyou Yuanming monastery 慈佑圓明寺 in Fuping 阜平. The monastery had halls of seven layers, one magnificent Amitâbha statue, and several qing of land. 31 After that, he returned to Mount Wutai. But it seems that it was not possible for him to stay long in a place of meditation; the governor of Shanxi province asked him to supervise the restoration of a bridge on a road near Taiyuan. At the same time, the Prince of Jin 晉32 asked him to rebuild the Yongming monastery 永明寺 in the city of Taiyuan. Miaofeng Fudeng changed the name into Yongzuo monastery 永祚寺 and added a new pagoda to the one that already existed. The temple is still popularly called Shuangta monastery 雙 塔 寺 (Two pagodas temple). He also built several new buildings including a large beamless hall on brick (Fig. 3). 33 When the buildings were completed, Miaofeng Fudeng received another request to repair a long bridge of 10 li to the south of Taiyuan. 34 The work had not been completed when Miaofeng fell ill, in the ninth month of 1612. 35 He returned to Mount Wutai and during his last weeks received numerous visits and prayers. Miaofeng Fudeng died on the nineteenth day of the twelfth month of the renzi 壬子 year during the fourth hour, which corresponds to 15 February 1613 at dawn36 . He was seventy-three years old. This description of Miaofeng Fundeng's activities from 1582 onwards bring to light two dimensions related to construction: one focused on the technical part of his work and the second related with his identity as a monk. The first is related to the particularly important number of brick constructions in Miaofeng Fudeng's work and more specifically of brick masonry vault (bridges, beamless halls, pagodas). This dimension therefore prompts me to consider the place of these constructions in the Chinese architectural system of the time. Was this knowledge widespread among craftsmen and if it was, why are there so few beamless halls in Chinese architecture? I will then address other types of architecture that have the same characteristics as the beamless hall and that can enlighten us on the knowledge of Miaofeng Fudeng. The second dimension is more related to the involvement of Miaofeng Fudeng as a Chan Buddhism monk in secular construction activities. His biography gives us several examples of bridge projects commissioned by important officials. In this regard, Miaofeng Fudeng belongs to an ancient tradition of religious expertise in the field of masonry structure.

A monk involved in vault architecture

Essentially of wood elements, classical Chinese architecture shows only very few examples of use of brick masonry and stone as supporting structure. The two techniques that enable the building in earth and wood are roughly consoles (dougong 斗栱) and pisé (banzhu 版築). The word tumu 土木 (earth and wood) means "building" in the common language. The finest and most remarkable innovations were made in the frames, roofs, consoles, lever arms, etc. Mortars, brick and stone walls are rarely mentioned in the sources mainly due to their marginal use in the "noble" architecture (l'architecture savante). In large buildings in the Forbidden City of Beijing, the wood frame is laid on wooden pillars that form the structure. Brick walls are not bearing the load but are only closures and partitions. In the popular architecture, wood frames were very numerous especially in Central and South China. In the north, where the lack of wood is constitutive for at least 500 years, a masonry architecture with most of the characteristics of the classical Chinese architecture (especially the house shape) has been developed.

2-1 The vault structure: a widespread knowledge

The masonry vault remains a rarity whose near absence in contemporary vernacular architecture as in the classical architecture is bound to be raised. However, the vault and the arch are not entirely absent, and have a very long existence in four types of building: tombs, bridges, arch openings (gates and windows) of pagodas and city gates. Here it is not a question of making an exhaustive history of the vault in Chinese architecture but of giving some historical points of reference in order to demonstrate the mastery of these techniques since ancient times. 37 During the Eastern Han, tombs involving arch or dome structures in bricks were widely distributed in China. At the beginning, the curvature of the barrel vault was relatively small. From the middle Eastern Han onwards stone tombs were very much in vogue. 38 The typical arrangement being a single-chambered construction, rectangular in plan, with walls faced with stone blocks. Arch construction, as a structural pattern, was adopted in brick and stone tombs. This practice continued well after the Sui and the Tang when most of the stone tombs were built in a square shape and a cupola-shaped roof. Between the latter Han and the early Ming, the barrel arch was widely adopted in aboveground construction, for example in the Ming tombs where the vault shape was the more common type. 39 Stone and brick can withstand very high pressure. The arch structure is the ideal form for stone construction. 40 As far as I know, the earliest records of the arch bridge are provided by the description on a molded brick made during Eastern Han dynasty 41 . One of the oldest remaining bridges, the famous Anji bridge 安濟 in Hebei which was built during the Sui period (589-618), presents a great segmental arch bridge consisting of a single arch, 37 meters in span which is in turn composed of 28 separated arches built up side by side in a series. 42 Like the building of an arch, the construction required only a narrow centering capable of being moved on as each arch was completed. The long practice in the Ming and Qing brought forth a new development in the technique of brick and arch construction. During this period, arched city gates of brick appeared, which were supposed to possess greater defensibility. Formerly before the Yuan, wood framework was used to carry load in the gateway and it was only from the 37 Bodolec, 2005, chaps. 1, 2 and3. 38 "Qin-Han muzang jianzhu", 1980, pp. 193-270. 39 Kuhn, 1996, p. 42 andpp. 53-54. Zhongguo tianye kaogu baogaoji, 1990, pp. 14-15. 40 Mao Yisheng, 1986. 41 Lü Pin, Zhou Dao, 1965, p. 17. 42 Liang Sicheng, 1936, pp. 1-16. southern Song to the Yuan that wood framework was gradually replaced by the barrel arch construction, for example the Heyi gate 和義 of Dadu capital of Yuan dynasty. 43 As we have seen in his biography, Miaofeng is known not only for his beamless halls but also for his stone bridges. John Kieschnick has shown with many examples, from at least the sixth century to the end of the Qing, that monks played a prominent role in building and maintaining Chinese bridges. 44 The sources most of the time only indicate the dates of construction, the place and the name of the monk in charge of the site. Rarely, technical details or the precise role of the monk in the building process is indicated. Some of them are known to have built tens or even hundreds of bridges, which suggests that they would have mastered some knowledge in architecture and labor management. 45 Bridges are also seen as acts of virtue and merit, as acts of compassion and building a bridge is valued by the Buddhist canon and the society. Miaofeng has undoubtedly benefited from the knowledge of the monks of the different monasteries where he lived, monks who probably had had the experience of such buildings.

It can therefore be said that Chinese artisans had mastered brick and stone masonry architecture since ancient times. The knowledge on mortars and bricklaying has allowed them to build quite sophisticated structures. The beamless halls attributed to Miaofeng Fudeng, in this sense, belong to the Chinese constructive system and he was probably able to lean on craftsmen familiars with the techniques of vault building. His expertise in bridges is proof of this. Nevertheless, a closer look at the vault structures permits to realize that the beamless halls use different construction techniques which can bring new elements to understand their rather scarce presence in Chinese architecture.

2-2 Vault with abutments: a technical innovation

Technically, the underground vaults for tomb, the bridges, the pagoda windows and the city gates have in common to be just "a hole in a mass", a vault which transmits its load (vertical force) and pressure (horizontal force) to a huge mass of earth (tombs, gates in the walls) or in the case of bridges, to bases on the ground. The forces which naturally cause the collapse of the vault are perfectly balanced thanks to these masses. These buildings belong to a same architectural environment. On the other hand, Ming beamless halls are built with extra-heavy walls to counterbalance the horizontal thrust of the barrel-arch construction. The emergence of buttress pilasters was an improvement, reducing the amount of brick needed and at the same time increasing the span of doors and windows and, of course, of the entire vaulted structure, enabling these to let in more light. 43 Zhang Yuhuan, 2000, pp. 176-177. Tushuo Beijing shi, 1999, pp. 217 and227. 44 [START_REF] Kieschnick | The Impact of Buddhism on Chinese material culture[END_REF]Kieschnick, , pp. 199-2008. 45 . 45 Kieschnick, 2003, p. 204. Mao Yisheng, 1986, p. 236. Needham et al. 1965, p. 160. At this time the vault, "which was just a hole in a massive structure, has been liberated to form a free structure above the ground." 46 The still extant beamless halls attributed to Miaofeng are of two types: large buildings (averaging 25 meters in length) and the smaller ones sometimes called in the documents "halls of flower" duodian 朵殿, which are smaller in size. The large structures exist at the Yongzuo monastery in Taiyuan, at the Xiantong monastery in Mount Wutai, but also at Wangu monastery where only some ruins remain. The structure of the Mount Emei monastery is quite different. The hall is unique in Miaofeng Fudeng's work with a square plan and a dome roof. On all four sides vaulted openings bring in light. At each corner of the roof and in the center of the dome are decorative Tibetan-style stupas. 47 Going back to the four buildings with important similarities, one might consider separately the interior and the exterior of the construction. While the outside aims to look like an ordinary Chinese temple, the inner structure is relatively complex, consisting of an elongated vault crossed by three or five oblong vaults. This helped to create a larger space in which Bodhisattva statues were installed. In the Yongzuo monastery, the vaulted buildings form the three sides of a square courtyard. The main building is 25 meters long and comprises five oblong rooms, each with a span of 3 meters. The elongated vault that crosses the oblong ones is more than 6 meters long. The layout of the other buildings in the courtyard is similar (Fig. 4). The main building of Xiantong monastery in Mount Wutai is 28.2 meters long and 16.2 meters wide with only three oblong vaults and an elongated transverse vault that is quite wide. At the rear of this building, one can observe the two constructions called duodian. As they can be called beamless hall because of their vault structure, they are significantly smaller. In a very interesting way, these two buildings are very close stylistically to those built at Mount Baoha and at the Yongzuo monastery in Taiyuan. The Mount Baohua gazetteer tells us the measurements: 10.24 meters high and 6.72 meters wide. 48 Their length is close to 8 meters. 49 Their vaults are elongated, which means that they are perpendicular to the main entrance. The vaults have a span of 3.50 meters. (Fig. 5) The beamless halls attributed to Miaofeng have remarkably shallow abutments: 3.5 to 4 meters thick for the largest vaults and 1.5 meter for the smaller ones. I did not find any Chinese source describing a method for calculating the thickness of the vault support walls, but according to a 17th century French book, a 6-meter barrel vault can be supported by an abutment of 1.5 meters. The calculations are not very complicated for skilled craftsmen but require a particular technicality. 50 There are no buildings of this style in later periods, although one exception is known: the Kaiyuan monastery 開 元 寺 in Suzhou (Jiangxi). Situated in the Ruiguan monastery 瑞光寺 in the southwestern quarter of the city, it is the only building remaining after many fires. 51 This beamless hall is attributed to a monk 48 The measurements are respectively 3 zhang 丈 2 chi 尺 and 2 zhang 1 chi. "Wenshu wuliang dian" and "Puxian wuliang dian," Baohua shan zhi, 1736-1795, juan 3, pp. 1b-2a. 49 Prip-Møller, 1937, p. 266. See also [START_REF] Jiang | Jiangsu de sanchu wuliang dian" 江蘇的三處無梁殿[END_REF], p. 87, fig. 2. 50 Adam, 1995, p. 182. Derand François, 1755, p. 10. 51 Demiéville, 1925, p. 245. Suzhou fuzhi 蘇州府志, 1829-1877, juan 39, pp. 9-10. Liu Dunzhen 劉敦楨, 1984, pp. 257-317. named Ruyuan 如 緣 and was built in 1618 according to the documents. 52 Its exterior style and inner structure are very similar to those built by Miaofeng, 53 which he most likely used as a model although no formal proof thereof can be provided (Figs 6 and7). As we have already said, very few buildings have these technical characteristics and can be called beamless halls. But the techniques of the arch shape for bridges, gateways of city walls and temples have been preserved during the Ming and Qing dynasties, and the techniques of the vault with abutments have been maintained until nowadays for vernacular architecture in Shanxi province and northern Shaanxi (so called Shaanbei). 52 Pan Zengyi 潘會沂, early XIXth, p. 13a. Niu Ruolin 牛若麟, Wang Huanru 王煥如, 1642, juan 24, p. 24a-29b. Wu Xiuzhi 吳秀之, 1933, juan 36, p. 6b-7b. Liu Dunzhen, 1984, p. 316, n. 24. 53 Jiang Shirong 江世榮 et al., 1955, pp. 85-86, fig. 1;and Boerschmann, 1925, p. 35, Abb. 10.

2-3 The vault with abutment for houses

Houses called yaodong 窑 洞 are located in the northern part of the loess tableland, in the basin of Huanghe, one of the most important alluvial deposit zones in the world. The golden colored loess is sometimes two hundred meters high. In this area a high concentration of cave dwellings dispatched on seven regions shelter nearly forty million people. In Shanxi, Shaanxi, and Gansu, more than fifty per cent of the inhabitants still live in yaodong54 . Those houses may be divided into three different groups. The tu yaodong 土窯洞 (only made in earth, deeply excavated in the cliff), the tianjing yaodong 天 井 窯 洞 (a type of dwellings of very spectacular effect because they are ten meters deep in a very flat landscape) and the duli yaodong 獨立窯洞 (independent type). The three groups have in common their name (yaodong) and also their plan, on average nine meters in length, four meters in width, three and half meters in height, and the use of the vault shape. The "independent type" (duli yaodong 獨 立 窯 洞 ) presents certain architectural and technical similarities with beamless halls (Fig. 8). Even though this architecture is less impressive and not intended for temple or monastery construction, there are numerous common elements: rectangular form, entry by the narrower side, bricklaying techniques, barrel vaults, etc. Could there be a connection between these vernacular constructions and Miaofeng Fudeng? Indeed, the biographies of the monk give us some clues. Miaofeng was born in 1540 at Pingyang 平陽 (modern Linfen 臨汾) in Shanxi province to a family named Xu 續. From the many episodes of his life before 1586, we learn that he had many opportunities to cross villages on his trips to the Xiyan 棲岩 Temple on Mount Zhongtiao 中條 山 55 or to Mount Jiexiu 介休山. 56 In the prefecture of Linfen, where Miaofeng was born and where he lived for several years, independent yaodong made of brick are still quite numerous today. The oldest among them were built for the local elite like the Shi family manor of Shijiagou 師家溝 in Fenxi 汾西 district which probably dates from the years 1760-1770. The independent yaodong for wealthy people were also built in cities, like the ones in Pingyao 平 遙 , as well as in rural areas, such as the huge manor of the Wang family in Jingsheng 静升, in Lingshi 靈石 district (Fig. 9).

Independent yaodong served not only as private houses but also as monasteries and temples in this region. This is the case for the Longtian miao 龍 天 廟 in the district of Fenyang 汾 陽 , Shanxi province, where epigraphs specified that the wood buildings and the yaodong were restored during the Qianlong era. 57 Sometimes these yaodong exist in the same places locations as the beamless halls, for example in the Yongzuo monastery in Taiyuan (Fig. 10). It is reasonable to suppose that, if independent yaodong were as numerous in Miaofeng Fudeng's time as in the late seventeenth century, he was familiar with the form as well as the techniques of construction. If we can refer to the current situation, the building of brick vaults requires many workers, laborers hired for the duration of the project. The only professionals are the master masons who understand the calculations for constructing the vaults (Fig. 11). Today, and most probably also in the Ming period, the most skilled ones were very much in demand and were engaged all around the district and sometimes beyond. 58 Fig. 9. Wang family manor, Jingsheng, Lingshi, Shanxi (©author) 55 Mountains in southeastern Yongji district. 56 Also known as Mian shan 綿山 in the district of Jiexiu 介休. 57 Lin Huiyin 林徽因 and Liang Sicheng 梁思成, 1934-35, pp. 13-14. 58 Bodolec, 2010, pp. 6-7. Like the building of the arches in the beamless hall and the yaodong, the construction of bridge arches required only a temporary structure to center the vault. The bricks were laid in the same manner, layer by layer from the springing point to the keystone (Fig. 12). One piece of evidence of this shared knowledge is found in Miaofeng Fudeng's biography. He was summoned several times to conceive other types of vaults than beamless halls, including the multi-arched bridges on the Wei River in Shaanxi and on the Yang River in Hebei. These bridges, constructed upon the request of civil officials, were great successes: much of Miaofeng Fudeng's fame came from bridge architecture. In a sense, this makes him part of the tradition of bridge building amongst Buddhist monks. But one still has to account for the uniqueness of his technique and style within this tradition. The assumption is made that Miaofeng, on the building sites in Sichuan or in Jiangxi, was able to presumably hire masons and bricklayers specializing in brick bridge construction. They could work on beamless structures and there was therefore no need to move a team of artisans from Shanxi province. This hypothesis is also corroborated by the architectural style of the beamless halls which borrow their decorative vocabulary from timber construction, but do so through carved brick pilasters, railings, and brackets. 59 Between the openings in the facade are brick pilasters set on molded pedestals. These pilasters have no capitals but decorative brackets instead, which support an architrave with three layers of brackets as a sort of cornice. All these elements are made of brick. The roofs also imitate the classical wood-temples: they are of saddle-back hipped roof xieshan wudting 歇山屋頂 following the roof style typically found in oneroom palace halls in Chinese architecture. The upper part is surmounted by a canopy. Miaofeng, familiar with the techniques of abutments was able to supervise their building and lead the artisans.

However, such technical skills are not sufficient to explain how Miaofeng successfully carried out these gigantic projects. Furthermore, we do not have much information about the cost of such achievements. If we follow Timothy Brook's indications, a full-scale reconstruction, which might take decades to complete, could cost some ten thousand taels; to restore or relocate an existing monastery would require two or three thousand. The cost of construction of one major building like a Buddha hall varied according to its size and magnificence, though the budget usually exceeded a thousand taels and could run as high as two thousand. 60

Miaofeng Fudeng and his Patronage

3-1 Local elite and high officials

As we have already seen, very early in his life Miaofeng received support from a member of the imperial clan, Zhu Junzha 朱俊柵 (r. 1558-1603), a fifth generation descendant of the founder of the Ming dynasty. Referred to as the as Prince Shen. Miaofeng asked him for the funds necessary to cast the first of the three statues, as well as the bronze pavilion for the monastery at Emei Mountain in Sichuan province. He agreed to finance that specific project and he accompanied Miaofeng to Jingzhou in Huguang, where the elements were cast. 61 In Taiyuan, circa 1610, Zhu Qiugui, known as the Prince Jin, asked him to repair the Yongzuo monastery.

These persons were members of the local elite, in Chinese: jin 衿, shen 紳 or even shi 士 , in other words people who can be defined as "a social elite that sustained its dominance through the mobilization of cultural and social, as well as economic and political, resources in the local context". 62 Local elites, who shared with the government the control of local affairs, represented an informal power in contrast to the formal power invested in the local government 63 . The Chinese gentry was the only group that could legitimately represent the local community in discussing local affairs with officials and in participating in the governing process. The gentry consisted of officials-active, retired, or even dismissed-including those who had purchased their official titles or ranks, and also included the holders of degrees or academic titles. Wealth or landed property per se was not a qualification for gentry status. Commoner-landlords did not belong to the gentry class, no matter how much land they owned. 64 In the case of a local district's public works, the role of the gentry was very important. Their members could finance a large part of infrastructure projects, such as the restoration of a city's walls or the construction of bridges. The construction of a bridge or a road was an opportunity to increase the prestige of the donor, and such constructions could be officially described as "acts of virtue" (yiju 義舉) or "good deeds" (shanju 善舉). 65 Indeed, Miaofeng was appointed to supervise the construction of bridges in various places in Shaanxi and Shanxi provinces. Those ultimately responsible were, however, high-ranking members of the imperial government. Among them were men such as Li Shida, the Minister of Justice; Wang Xiangqian, governor of the North Metropolitan Province and later governor of Sichuan province; or the governor of Shanxi province, whose name does not appear in the sources. It is unclear whether these officials summoned Miaofeng to build bridges within the framework of an official request or as the result of a benevolent action by the local gentry. In the case of Li Shida, for example, who was a native of Jingyang district in Shaanxi, near the Wei River where Miaofeng erected a bridge, we can argue that this represented a gentrysponsored project. But for Wang Xiangqian, a native of Shandong province, the connections are more tenuous.

It is in relation to beamless halls and statues of Bodhisattvas that the role of the gentry in Miaofeng Fudeng's activity was more obvious. In the case noted 61 Hanshan Deqing, in Xu Zangjing, juan 73, no. 1456, p. 675. 62 Brook, 1993, p. 15. 63 Ch'ü Tung-ts'u, 1962, p. 168. 64 Ibid., pp. 170-172. 65 Yang Liansheng, 1964, p. 13. Kieschnick, 2003, pp. 208-212. above, Wang Chonggu, a native of Puzhou in Shanxi, chairman of the Ministry of War, asked Miaofeng to restore the pagoda and the main building of the Wangu monastery, where Miaofeng Fudeng had lived as a monk in his youth. Another important magistrate, Wang Jiyu, Grand Coordinator of Sichuan province, agreed to patronize the Wannian monastery on Mount Emei in circa 1600. These names are not the only examples of the involvement of officials in the restoration and building of Buddhist temples. Although they were educated in Confucianism, the local elite of late Ming period had chosen to patronize Buddhism, and particularly Buddhist monasteries, instead of other forms of benevolent deeds. As Timothy Brook demonstrated, "monastic patronage became significant in the late Ming because it constituted a local forum for the autonomous organization of the gentry."66 Buddhism provided a context for social elites to create an identity for themselves in terms that were to some measure independent of the state. The individual itinerary of Miaofeng is another example of this fortunate situation. Like other monks, he was able to benefit from opportunities to spread Buddhism and to develop the monasteries. The fact remains that the most important sponsor of Miaofeng was at the top of the social pyramid in the person of the Empress Dowager, mother of the Wanli emperor. How could a simple monk get such support? Let Hanshan Deqing explain how it happened.

3-2 Miaofeng Fudeng's Prestigious Patron: the Empress Dowager

In 1571-1572, the Prince of Shanyin, Miaofeng Fudeng's protector and mentor, had just built a monastery at Mount Nan 南,67 and he invited Miaofeng to reside there. Some time later, the Prince wanted to obtain a copy of the Yongle beizang 永樂北藏 (the so-called Northern version of the Tripitaka) for the new monastery. He did Miaofeng the great honor of asking him to carry out this task. Being the owner of one of these sets of the Buddhist canon was not a trivial privilege, and calculations show that only fifteen sets of the Tripitaka were given away by the throne in the 1590s. 68 Miaofeng met up with Hanshan Deqing again when he arrived in Beijing in winter of 1572, and their friendship grew stronger. In 1573, the first year of Wanli reign, Miaofeng obtained a copy of the Yongle beizang; when he returned to Puzhou, Hanshan Deqing accompanied him. The Prince of Shanyin was particularly pleased and agreed to let Miaofeng go to Mount Wutai. At the end of February 1574, 69 Miaofeng and Hanshan Deqing started out towards the sacred mountain, nearly 700 km to the North. The route ran through Pingyang, where Miaofeng was born. As his parents had died unburied several years earlier, he decided to perform the appropriate rites. Arriving at Mount Wutai, the two took up residence in Longmen 龍門, on the Beitai 北臺, for three years. [START_REF]The Mount Wutai gazetteer adds details on the exact place of their stay: the Miaode an 妙德 庵, where a temple still exists today. "Miaofeng dashi zhuan[END_REF] During all this time, they continued to be anxious about the after-life of their respective parents, and they decided to make a copy of the Huayan jing, the Flower Garland Sutra. Hanshan Deqing decided to use ink made of blood and gold flakes, while Miaofeng chose instead the blood of his own tongue mixed with red ink. [START_REF]The Mount Wutai gazetteer even specifies that the gold flakes and the paper were provided by the Empress Dowager herself. Frequent references to the practice of blood writing appear from throughout Chinese Buddhist history in biographies of monks[END_REF] We assume that they began to copy the sutra in 1577.

When the copies were completed, Miaofeng wanted to hold an Undiscriminating Great Assembly (wuzhe dahui 無遮大會, Sanskrit, pañcavarsikāparisad), which was a meeting of religious figures and the faithful for confession and forgiveness of sins. During a whole year, Miaofeng Fudeng raised funds and support, and finally, in December 1581, the ceremony was ready to be held. The presentation of the copied sutras and the preaching took place in the Dayuan temple, newly restored thanks to donations from the Empress Dowager. Originally, this restoration in which Miaofeng Fudeng seems to have had no implication involvement, was planned in memory of the late emperor and for the reiging of the new emperor Wanli. The Empress Dowager sent artisans to help complete the work along with officials on behalf of the outer court (Ministry of Works) and from the inner court (eunuchs actually). [START_REF]For a complete depiction of the Wuzhe dahui see Zhang Dewei[END_REF] The ceremony lasted 120 days, 73 and it is said that Miaofeng Fudeng invited 500 eminent monks to participate. In his autobiography, Hanshan Deqing described the ritual and said that there were no less than ten thousand people who came from all directions. [START_REF] Deqing | [END_REF] While the ceremony was already underway, messengers from the Empress Dowager came to Mount Wutai to pray for the birth of a crown prince. This was part of a major effort to help Wanli Emperor, who had married three years before, have a son. Hanshan Deqing suggested incorporating the prayer in the ceremony but both the envoys and Miaofeng Fudeng objected mainly not to interfere with Wanli Emperor own messengers sent to Mount Wudang 武 當 山 , a sacred Daoist mountain in the northern part of Junxian 均縣 in Hubei province. The Empress Dowager's request was thus unofficial but in the end the two monks accepted to change the object of the ceremony and pray for the birth of an heir. In the eighth month of Wanli 10 (1582), Zhu Changluo 朱 常洛 (1582-1620) Wanli's first son, was born and it seems that the credit counted/was given? to the Mount Wutai ceremony and personally to Miaofeng Fudeng and Hanshan Deqing. 75 Right after the end of the Wuzhe dahui, the two friends departed from Mount Wutai and separated. As Hanshan Deqing went to Zhengding 正定 (Hebei) to convalesce, 76 Miaofeng Fudeng went to Mount Luya in northern Shanxi province to retire to a small hermitage (an 庵). 77 After the birth of the heir apparent at the end of 1582, the Empress Dowager sought to express her gratitude to the two monks. As mentioned above, when she learned of Miaofeng Fudeng's retreat, she ordered him to restore the main temple of Luya Mountain, the Huayan monastery, and give funds to build an iron pagoda of seven stories, the Wanfo pagoda 萬 佛 铁 塔 , on the top of the mountain. Miaofeng Fudeng, at the age of forty-two, thus began a new carrier as an architect. 78 Imperial benevolence towards him continued throughout the rest of his life. When he died, in February 1613, the imperial family contributed lavish gifts for his funeral; the Empress Dowager herself sent 1,000 jin 金 and 500 pieces of cloth. The Emperor ordered that a tomb pagoda be erected for him in the western part of the Xiantong Monastery. It is recorded that an inscription calling him zhenzheng Fozi 真正佛子, "true disciple of Buddha," was carved. 79 The emperor also ordered the completion of all the buildings in progress at the time. 80 The biography written by Hanshan Deqing emphasized the close relationship between Miaofeng Fudeng and the Empress Dowager Cisheng. In doing so, he indicated that the ritual organized in 1582 proved to be a key event in Miaofeng Fudeng's life. As in his autobiography, he stressed the correlation between this event -which took place in spite of strong objections from his contemporaries -and the birth of the crown prince ten months later. Without entering into details, we have to keep in mind that Hanshan Deqing himself had had a special relationship with the Empress Dowager. Specifically, he wanted to attract her attention to his personal project: the restoration of the Dabao'en monastery in Nanjing, which was severely damaged by fire in 1566. The insistence on incorporating the two events could be a way to establish a link with the imperial court, especially with the Empress Dowager. Miaofeng Fudeng, sometimes reluctantly, according to the sources, also benefited from these circumstances. 75 This son was the subject of a major dispute between Wanli Emperor and his mother. Zhu Changluo under the reign name of Taichang 泰昌 only reigned for one month in 1620. Zhang [START_REF] Dewei | A Fragile Revival: Chinese Buddhism Under the Political Shadow, 1522-1620[END_REF], chap. 3, p.73 sq. 76 Hsu Sung-peng, 1978, pp. 74-75. 77 District of Kelan 苛嵐縣. 78 These constructions were destroyed during the battles that brought the Ming dynasty to an end; see the gazetteer of Shanxi province, quoted by Glahn, 1976, p. 463. 79 Mount Wutai stele. Su Weilin (jinshi 1598) text copied in Zhao Lin'en, 2005, p. 341. Zhang Dewei, 2016, p.328, note 6. 80 "Miaofeng dashi mu", Qingliang shan zhi, 1887, juan 3, p. 30b.

Hanshan Deqing, more directly involved than Miaofeng Fudeng in power struggles within the Imperial court, was less fortunate. 81 He was arrested in 1595 and exiled to Canton a few months later. Indeed, at this moment, a court controversy centering on the issue of succession occurred: the Empress Dowager and her son, the Wanli emperor, were the leaders of two opposing factions. During this court dispute, religion, especially Buddhism, was widely used as both a protective shield and a weapon. As a consequence, we can see some effect in the support of temple construction when, for example, the Empress Dowager's party was the weaker, in the years 1595 to 1602.

However, the Empress Dowager Cisheng was the most powerful woman in the religious life of the court, in the capital, and in the empire. The number of her patronage projects and the sums she provided were enormous. Miaofeng Fudeng was not the only Chan monk she protected and financed. In the year 1602 alone, two other monasteries were restored with funds from the Empress Dowager. The Yuquan Monastery 玉泉 寺 in Dangyang district 當陽 (in modern Hubei province), underwent a major restoration led by the Abbot Wuji Zhenghui 無跡正誨 (fl.1602), a monk of empire-wide reputation. The project was initiated with the help of imperial patronage, when Empress Dowager Cisheng gave Zhenghui a grant of 10,000 taels to carry out the work. She also presented him with a copy of the imperially printed Tripitaka in the name of the Wanli emperor. 82 In the same year, another monk, named Xinkong Mingkai 心空明開 (1568-1641), went to Beijing precisely to procure a copy of this document. Knowing that the Empress Dowager had long suffered from an incurable eye affliction, Mingkai presented to the throne a medical lotion, which proved to be effective in easing her condition. The Empress Dowager showed Mingkai her gratitude by having the emperor present not only a copy of the Tripitaka but also 1,000 taels of silver. The imperial donation stimulated a spate of contributions from provincial, prefecture and district officials, as well as from eight prominent members of the Zhucheng 諸 城 district gentry. Within five years, the Guangming Monastery 光 明 寺 was completed. Its library housed the 6,780 volumes of the Tripitaka, which were delivered to the monastery in 1607. 83 The Empress Dowager seems to have financially supported at least forty-seven Buddhist temples, from the foundation of institutions to renovation, rebuilding, and erecting statues. 84 She also had close links with monks, as we saw in the case of Miaofeng Fudeng. She was central to most of his projects; perhaps she even was at the source of his building activity as an architect.

Conclusion

Miaofeng Fudeng's individual renown as a leading monk responsible for a number of important construction projects involving the beamless form is unquestionable. I would argue that the period and historical circumstances in which he lived were favorable to Miaofeng Fudeng's work. He lived in a very particular time of the Ming dynasty, the Wanli era, when Buddhism flourished more than in other periods. The late Ming was a period of revival for religious institutions, when Buddhist monasteries were restored all over the empire. Sometimes the abbot received financial assistance or moral patronage from the emperor himself or from the members of his family such as the Empress Dowager. In other cases, the funding came from civil servants in the provinces or from influential eunuchs. The temples were then said to be gongjian 公建 (built by officials); this had a moral connotation and could encourage members of the elite to contribute donations to the work in progress. 85 Although restrictions on building temples were officially maintained, they were often not enforced. 86 The Ministry of Rites had a say in the appointment of monastery officials and abbots of important temples. Thus, Miaofeng Fudeng was promoted to the status of abbot zhuchi 住持 of the Xiantong Monastery on Mount Wutai. He contributed to its development by building a beamless hall there. 87 As Timothy Brook has shown, the chief context of this revival was the formation and expansion of the local gentry. 88 It seems that one can identify two periods of greater activity in Buddhist construction during the Ming period. The first was at the beginning of the dynasty, during the Hongwu 洪 武 (1368-1398) and Yongle 永 樂 (1403-1424) eras, which correspond to the phase of rebuilding the empire after the reconquest. A second period opened during the Wanli reign (1573-1620), and continued until the beginning of the Qing dynasty in 1644. 89 The beamless halls attributed to Miaofeng Fudeng were all built during this period.

However, it was more difficult to understand why all Miaofeng Fudeng's constructions belonged to the field of masonry architecture and rather than that of classical wood architecture-why his expertise lay in the knowledge of the forces governing vaulted structures. As we learned in his biography, he was summoned several times to conceive types of vaults other than beamless halls, for example, the multi-arched bridges on the Wei River in Shaanxi and the Yang River in Hebei. These bridges were commissioned by civil officials and were recognized as great achievements. Much of Miaofeng Fudeng's fame came from the bridge architecture. But Miaofeng Fudeng's work on brick-vaulted temple structures is impossible to dissociate from his origins in Shanxi, where brick vaults were relatively common in vernacular architecture, and which were un-

Fig. 1 .Fig. 2 .

 12 Fig. 1. Xiantong monastery, Mount Wutai, Shanxi, 1606 (©author)

Fig. 3 .

 3 Fig. 3. Yongzuo monastery, Taiyuan, Shanxi, 1612 (©author)

Fig. 4 .

 4 Fig. 4. Yongzuo monastery, Taiyuan, Shanxi, 1612 (after Liang Ssu-Ch'eng and Fairbank, 1983, p. 173)

Fig. 5 .

 5 Fig. 5. Longchang monastery, Mount Baohua, Jiangsu, 1605 (after Prip-Møller, 1937, p. 266)

Fig. 6 .

 6 Fig. 6. Kaiyuan monastery, Suzhou, Jiangsu, 1618 (Chinese postcard around 1920)

Fig. 8 .

 8 Fig. 8. Jiang family manor, Liujia mao, Mizhi, Shaanxi (©author)

Fig. 10 .Fig. 11 .

 1011 Fig. 10. One of the lateral temples, Yongzuo monastery, Taiyuan (©author)

Fig. 12 .

 12 Fig. 12. Wannian bridge 萬年橋, Nancheng, Jiangxi, 1634-1647 (after Liu Dunzhen, 1933, fig. 10).

Hanshan Deqing, in Xuzang jing, p. 676, upper part, line 15. See Glahn, 1976, p. 465.

Zhang Dewei, 2016, p. 353.

This was probably Zhu Qiugui 朱求桂, Prince of Jin 晉 (fl. 1611-1630), even though the records say that he officially became Prince of Jin only in 1613. His father died in 1610.

Hanshan Deqing, in Xucan jing, p. 677, upper part, line 23. Weidner, 2001, pp. 131-132.

A li was then equivalent to approximately 576 metres.

Wanli renzi year, autumn, ninth month 萬曆壬子秋九月, equivalent to the end of September to October 1612.

萬曆壬子臘月十九日卯時. Hanshan Deqing, in Xucan jing, p. 677, middle part, ligne 2.

Translated from Adam, 1995, p. 180.

As far as I know, the sources do not give information about the particular form of this building. See Millbank, 2014, p. 18 for a more detailed description.

Zizhou xianzhi 1993, p. 267; Zichang xianzhi 1993, p. 440; Lingshi xianzhi 1992, p.193; Mizhi xianzhi 1993, p. 257.

Prince of Shanyin in historical sources, he played an important role in the first period of Miaofeng Fudeng's life. He encouraged him to further his knowledge of Buddhist texts and travel around the empire. The influence of this prince was certainly decisive for the architectural education of the monk, since he had had the opportunity to visit many monasteries. Miaofeng crossed the paths of two

Brook, 1993, p. 320.

There are at least eight places named Nanshan in Shanxi province; I assume this was the place nearest to the residence of the Prince of Shanyin, at the Wangu monastery, Yongji, Yuncheng district.

Hsu Sung-peng, 2000, pp. 95-96.

Wanli jixu year, spring, first month 萬曆甲戌春正月, equivalent to the end of February to the beginning of March 1574.

Zhang Dewei, 2014, pp. 263-285.

See the Yuquan shan sizhi (1694), juan 1, p. 5a; and the Dangyang xianzhi (1866), juan13, p. 35b, quoted in Brook, 1993, p. 291, n. 30.

Wuliang shanzhi, juan 2, pp. 1a, 4a-b, 14a-16b; quoted in Brook, 1993, p. 241, n. 46.

Zhang Dewei, 2000, pp.

85-86.

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