domain: phys sdu

text:
Brian Craig
email: bic5021a1956@gmail.com

Elementary properties of colour charge and the fine structure constant as revealed by the dynamics of the P-sphere

Keywords: fine structure constant, dyon, Euclidean space, magnetic charge, colour charge

This paper explores the formation of Three-Dimensional Euclidean space and some of the fundamental characteristics of Coulomb charge and Colour charge. A periodic lattice of sites, with each site being the centre of a ball (P-sphere) possessing charge, angular momentum and tachyon-like mass, at least P+1 dimensions, is shown to provide a good model for a short-lived 3-dimensional Euclidean space with NaCl like cubic symmetry. Of particular interest is the derivation of the value for the fine structure constant , finds each ball to be a hollow hypersphere and possessing the charge of a dyon, containing both magnetic charge (26 units of Dirac monopole charge, g) and Coulomb charge (e/16). The calculated value for  is reproduced to the accuracy of the measured value (1/137.035999206). Either a simple dipole of magnetic charge or a quadrupole of magnetic charge may be a precursor for this lattice. This (NaCl-like) virtual state is proposed to transform to another NaCl lattice where the lattice sites are occupied by dyons of magnetic charge of single unit Dirac monopole, paired with Coulomb charge. Half of the sites have (magnitude) Coulomb charge of 27e, and half have charge 27⅓e. This state gives rise to interesting types of excitations of magnitude ⅔e and ⅓e. Coulomb charge and three Colour charge are the four dimensions of a 3-sphere.

Introduction

The aim of this paper to present qualities of the state obtained from the derivation of the value for the finestructure constant, in the context of the initial formation of a three-dimensional space from an apparent void, possessing a total energy of zero. A subsequent phase change leads to release free energy in the form of excitations which are assumed to possess qualities of Coulomb and Colour charge. This paper builds on two earlier papers [1,2] which introduced the basic concepts and recent work has found a key relevant state. The state is a lattice of dyon charge, possessing five additional compact dimensions. A model for colour charge and coloured bosons is deduced for this (3+5)-dimensional space, including the Coulomb values of -1/3 and +2/3. The five compacted dimensions are assumed to relate to one for Coulomb charge (such as the electron -e), one for magnetic charge (g) and three colour charges (A, B and C). Euclidean space is described as a lattice of touching P-spheres, which are hollow in (P-1) dimensions. The angular momentum within these spheres is critical. The magnitude of charge is dictated by the Kaluza-Klein approach (Lorentz force) relating charge with speed in a compact dimension.

Rather than just the three-dimensional lattice appearing suddenly at time zero, this paper also looks at the scenario that a finite-sized multiple, such as dipole, quadrupole or octupole, makes the initial appearance and the lattice is built from a replication of this multipole, as an effective unit cell. In particular the octupole of a square of two positive charges in opposing diagonal and two negative charges of the other diagonal can be replicated (many times) to first form the cubic NaCl-like lattice. Subsequent phase occur until the magnetic charge is a single unit (Dirac monopole, g). The approach needs to consider Lorentz symmetry of the observable Universe, and yet introduces the notion of crystalline symmetries for space as its own entity. To proceed in this task, we introduce an order parameter  where =0 is Lorentz symmetry and crystalline symmetries can occur for ≠0. To succeed, we need all free particles, those which can exist in isolation, to have no measurable interaction with the proposed crystalline structure of space for the range of energy that have been used by experiments to date. Effective Lorentz symmetry is aided by the underlying structure continuously oscillating between similar, but different, structures.

The approach is based on a variation of the tachyon field method, where the sombrero is turned upside down by reversing the sign of the coefficients as well as including the sixth order, to give 𝑉(𝜑) = - +  𝜑 - 𝜑 +  𝜑 where o , 1 , 2 , 3 are all positive, and  is a scaler parameter. The system has Lorentz symmetry at = 0, and this Lorentz invariant system is stable. The global minimum is =0 where for small , V(0±)= -o+r 2  2 , and r is a real mass.

However, this Lorentz-symmetric state is not the initial state. Instead, the initial state is assumed to have zero energy, does not possess Lorentz symmetry, is a local maximum, where dV()/d=0 at =T. This occurs for T, and V(T)=0 for zero total energy, giving

 =  3 𝛬  -  3 𝛬 1 - 1 3 𝛬
where

𝛬 = 1 -1 -3   
This value T will correspond to a local maxima, where V(T ± )= -i 2  2 , and i is the imaginary tachyon mass and

𝜑 =  3 1 -1 -3   
indicating a virtual state possessing zero total energy, and a finite life-time t , ≈ h/(ic 2 ). These states described quantised 3D space. Each point is a centre of a charged dyon, possessing both a magnetic charge jg and a Coulomb (electronic) charge ne/j, for the rational fraction n/j. The charge -e corresponds to the electron charge, and the charge g is the corresponding Dirac magnetic monopole charge. Each dyon has a mass containing an imaginary part.

The key feature is that this Tachyon state T is the state prior to the big bang and is quantised as a crystal lattice.

This state has total energy V(T)=0, and will not possess Lorentz symmetry as this state is not anisotropic. Apparent Lorentz symmetry is achieved for excitations of the Coulomb component of the dyon charge following a phase change. The ground state describing the isotropic crystalline state for 3D space (T) is not a single crystalline structure, but a resonance between two states, differing in Coulomb charge.

Earlier versions of this work have been placed on the web [1,2] had incorrectly deduced a large value for the magnetic charge for the final Euclidean state. Instead, such states are intermediate states and the final state has much lower values for the magnetic charge. (This preliminary works is not well written by myself with of several typographical errors and clumsy explanations.) Recent work has focussed on tracing the transition from an original multipole through a series of three-dimensional structures. The multipole is the original state, which can be thought as a small seed crystal, somehow induces replication to form the current three-dimensional structure for space. Attention is given to the formation of colour charge, bosons that exchange colour charge and an explanation for the puzzling Coulomb charge components of magnitude e/3 and 2e/3 and e.

Underlying Concepts for the Tachyon State

The virtual tachyon (=T) state

First of all, some simple underlying assumptions are made for the virtual tachyon (=T) state. The fundamental object of space is assumed to be the P-sphere, a spatial object of radius r and possessing P angles. This object has P+1 spatial dimensions and a hypersurface of P spatial dimensions. Three-dimensional space is assumed to be a network of touching P-spheres. A simple model for an extended object is a periodic lattice of touching spheres. The charge of the sphere is assumed to be at the centre of the sphere, and the quantum rigid rotor model is used for the angular momentum of the sphere, where mass is located on the hypersurface. The mass of each lattice point is m=mr+imi. The ratio of mr/mi is a key quality. The tachyon is a state with this ratio much less than unity. A stable state is where the ratio is much higher than unity. The absolute values of mr and mi are not known.

If P=2 then the touching spheres of the three-dimensional lattice are all hollow. If P>2 then the mass is distributed through the three-dimensional sphere and there are compactified dimensions. As the spheres touch in the three dimensional lattice, and if r describes both the radius of the P-sphere (P>2, r 2 =x 2 +y 2 +z 2 +w 2 +..) and the distance from the sphere centre to the touching point (r 2 =xTP 2 +yTP 2 +zTP 2 ), where (xTP,yTP,zTP,wTP,..) are the coordinates of the touching point, then the magnitude of any compactified dimensions (wTP) at the touching point needs to be zero. The model does assume r is the same, and the compactified dimensions are zero at the touching points. Compactified dimensions are relevant as the Kalusa-Klein concepts relate the speed of a mass (of a parton) in a compact dimension to the value of charge (of the parton).

The T lattice has aether-like qualities rather than Lorentz symmetry. Each ball is a (P=N+1)-sphere (S N+1 ) possessing angular quantum number L [1], radius r, and dyonic charge q=(ne/j, jg) where -e is the electron charge and g is the Dirac magnetic charge, with integers n and j. The Madelung constant (M) of the lattice structure (position vectors ri), given by

𝑀 = 𝑒 𝑒 𝑒 𝑟 𝒓 -𝒓 exp - 𝒓 -𝒓 𝑟 𝑜
where the term (ro -1 =mic/ħ) is an attenuation factor, ħ is the reduced Planck constant, c the velocity of light. The distance rac=2r is that between nearest neighbours, possessing Coulomb charge ei (and magnetic charge gi) with the same magnitude but opposite sign, and for ro >> rac, the value of M is very close to the solid state value calculated for an infinite crystal. In fact no significant attenuation is found for the proposed state for T as the solid state value gave complete agreement. An analogy of this 3-dimensional Euclidean space is a salt crystal where rac is the distance between a cation and an adjacent anion.

We also consider a small finite sized multipole, such as a dipole (one positive and one negative, a quadrupole (two positive and negative in a square) of octupole (four positive and negative in a square) then the summation is over a small number of sites to produce an effective M. If the value for rac is not many orders smaller than r0, due to all the mass distributed to a handful of poles, then M may be in need of a small reduction, by the fraction (1-rac/r) where rac/r <<1. The multipole may be the initial structure, acting as a seed for crystal growth.

Fine structure Constant

The value for  is given by the following equation 𝛼 = 1 4𝜋𝜀 𝑒 ħ𝑐 which is the Coulomb expression for  for electronic charge, where the (Coulomb) charge of an electron is -e. On the other hand assuming the charge unit q is instead the Dirac magnetic monopole, possessing charge g, defined as

𝑔 = 𝑘 2𝛼 𝑒𝑐
where the Dirac normalisation condition require k to an integer, then for k=1, (4𝛼) = ħ𝑐 𝜇 4𝜋 𝑔 which is the magnetic charge expression for . The aim of this work is to derive an expression for ħc in terms of Kq 2 , where q is the charge unit and K is a constant, and thereby gain an expression for .

The third type of particle is the dyon, possessing both electronic charge and magnetic charge. The interaction between two dyons with respective charge (e1,g1) and (e2,g2) sees the Potential energy is of the form [START_REF] Chuu | Path integral quantization of the relativistic dyonium system[END_REF] 𝑉(𝑟) = 𝑒 𝑒 4𝜋𝜀 𝑟 + 𝜇 4𝜋

𝑔 𝑔 𝑟 and magnetic potential vector A(r) for r =(x1,x2,x3) is

𝑨(𝒓) = ħ𝑄 𝑥 𝒙 𝟐 -𝑥 𝒙 𝟏 |𝒓|𝑥 𝑥
where x⊥=(x1,x2,0) and Q=e1g2-e2g1. The parameter Q = 0 when e1 = -e2 = qee, and g1 = -g2 = jg for n, which satisfies the Dirac condition for magnetic charge. The value for the lattice energy ELE is given by the expression

𝐸 = ½ 𝑀 𝑟 𝑞 4𝜋𝜀 + 𝜇 4𝜋 𝑗 𝑔 = ½ 𝑀 𝑟 𝑞 + 𝑗 4𝛼 ħ𝑐𝛼
where M is the Madelung constant for the lattice array (of positive and negatively charged dyons). Note the dyon charge (q) is Coulomb charge (qee) plus magnetic charge (jg).

The Derivation of the expression for the fine structure constant from the Dirac Expression

Each element of three-dimensional array has the same value of tachyon-like mass-energy (m = mr + imi). A tachyon possesses a non-zero imaginary part of the mass-energy and such quantum states may have a virtual or transient existance. The lattice energy due to lattice interaction between these charges, the mass-energy, the rotational energy and gravitational interaction are the same for each site and the summation of all these energy components is assumed to be zero. The interactions within the tachyon state are limited to an extended although finite range, as dictated by mi. The total energy per site is given by Dirac expression 𝐸 = ((𝑝𝑐) + (𝑚 𝑐 ) ) / + 𝑉 where the potential energy is the sum V=ELE+EG of the lattice energy ELE and the Gravitational energy EG. At this stage no assumption is made on the nature of q or K, giving

𝐸 = ½ 𝐾 𝑟 𝑞 𝑀
where M is always negative. The Madelung constants calculated for salt crystals [START_REF] Tavernier | Clifford boundary conditions: a simple direct-sum evaluation of Madelung constants[END_REF][START_REF] Zucker | Madelung constants and lattice sums for hexagonal crystals[END_REF] provide good estimates when 𝑟 ≫ 𝑟 . Otherwise an effective Madelung constant could be calculated as a convergent finite summation, arising from significant exponential attenuation when ro and rac(=2r) of comparable magnitude. For the dyon then Kq 2 =e  ((qe) 2 +(j/2) -2 )/4o.

Gravitational terms

Newtonian gravity is assumed as the rigid lattice is uniform and all positions are fixed, and the term is calculated over an extended volume encompassing a very large number of lattice positions.

In extended three dimensional space (3D), the gravitational energy for the transient state having a mass-energy of m is

𝐸 , (𝑚) = - 1 2 𝜌 exp (- 𝑥 𝑟 )𝑢(𝑥)4𝜋𝑥 𝑑𝑥
where D is the mass-energy density, with ro -1 =mic/h. The mass-energy density  is derived from the particle mass energy m, is given by the expression 𝜌 = 𝐷 𝑚 𝑟 where D accounts for packing of the touching spheres within the lattice, where D=for the NaCl structure and D=   for the CsCl structure. Newtonian gravity uses 𝑢(𝑥) = 𝐺 , giving

𝐸 (𝑚) = -2𝜋 𝑚 𝐺𝐷𝑟 𝑟 = -𝑉 , 𝑟 𝑚 𝑚 + 𝑖
where

𝑉 , = 2𝐺𝜋𝐷 ħ 𝑐 𝐸 , (𝑚) = -𝑉 , 𝑟 𝑚 𝑚 + 𝑖
noting VG,PD must be positive, and P is the number of uncompacted (spatially extended) dimensions.

3.2 Energy due to rotation in N+1 sphere.

The spatial quanta are assumed to be an (N+1)-sphere (P=N+1) of radius r = rac/2, which implies this model Euclidean space to be a system of touching spheres. The momentum squared p 2 , where there is only angular momentum within an (N+1)-sphere, which is given by the expression [START_REF] Frye | Spherical Harmonics in p Dimensions[END_REF]

〈𝑝 〉 = 1 𝑟 〈𝐿 〉 = ħ 𝑟 𝐿(𝐿 + 𝑁)
for a rigid rotor of the radius r and angular quantum number L, which is an integer. This is the equation for a rotation about a point, which is an axis of zero dimensions. If the axis has one dimension then we have a hypercylinder with the (N+1)-sphere as the "hyper-circle" in (N+2)-dimensional space. The above equation describes this motion. To make a simple example, consider three-dimensional space. A mass rotating in a 1-sphere (circle) around a line (1-dimesional axis) in 3-dimensional space, (which is a hollow cyclinder). This uses the above expresion for N=0. A 2-sphere around a point in 3-dimensional space uses the above expression for N=1.

This will be generalised to be a (N+1) sphere rotating about a axis. The axis may have zero or more dimensions. Such dimensions of the axis will be described as passive. The dimensions of the (N+1) sphere are described as active. Motion changes within those active dimensions. The above equation describes this rotation.

The expression for the fine structure constant

The total energy E is assumed to be zero, describing a transient state which does not require either an input or a release of energy, giving (𝑝𝑐) + (𝑚𝑐 ) = (𝐸 + 𝐸 (𝑚)) which, on substitution becomes

(ħ𝑐) 𝑟 𝐿(𝐿 + 𝑁) + (𝑚𝑐 ) = (𝐾𝑀𝑞 ) 16𝑟 - 𝐾𝑞 2𝑟 𝑀𝑉 𝑚 𝑚 + 𝑖 + 𝑉 𝑟 𝑚 𝑚 + 𝑖
The transient state is assumed to have a value for mi that allows ro>>r. The value for mi cannot be zero as we need an imaginary contribution to the above equation, as will be shown below. However mi can have an extremely small but positive value and in such a case the transient state may become very long lived.

Multiplying both sides of the previous expression by r 6 Then if  is the of unity, mr the same order or less than mi, then mi is the order of the Planck mass (2x10 -8 kg).

The 2-torus, or square lattice, (P=2) gives from which the solution is

𝑉 , 𝑟 (𝑚 𝑐 ) = 𝐾𝑞 𝑀 2(𝑚 𝑐 ) 2 𝑚 𝑚 -1 1 ± 1 + 16 𝑚 𝑐 𝑟 𝐾𝑞 𝑀 𝑚 𝑚 -1 giving [𝑉 ] ± = - 𝐾𝑞 (-𝑀)𝑟 4 𝑚 𝑚 -1 1 ± 1 + 16 𝑚 𝑐 𝑟 𝐾𝑞 𝑀 𝑚 𝑚 -1
where VGPD must not be negative, noting that M<0. If mr>mi then the term outside the bracket is negative, requires the the minus sign, as below

𝑉 (𝑚 > 𝑚 ) = - 𝐾𝑞 (-𝑀)𝑟 4 𝑚 𝑚 -1 1 -1 + 16 𝑚 𝑐 𝑟 𝐾𝑞 𝑀 𝑚 𝑚 -1
If mi>mr then the term outside the bracket is positive, then requires the term inside the square root of the solution

𝑉 ± (𝑚 < 𝑚 ) = 𝐾𝑞 (-𝑀)𝑟 4 1 - 𝑚 𝑚 1 ± 1 -16 𝑚 𝑐 𝑟 𝐾𝑞 𝑀 1 - 𝑚 𝑚 to satisfy 1 -16 𝑚 𝑐 𝑟 𝐾𝑞 𝑀 1 - 𝑚 𝑚 > 0
Now for mrc 2 <<Kq 2 , we have noting the term in the square brackets should be extremely small for magnetic charge (j>0), and r is extremely small (r<<ro). It is important to note this is the tachyon case, when mi>mr , and allows for short-lived transitional states.

𝑉 ± (𝑚 < 𝑚 ) = 2 𝑟 𝐾𝑞 (-𝑀) 𝑚 𝑐 𝑟
The non-tachyon case (mi<<mr) is a potentially stable state, and this is a requirement for the Euclidean space. We define the residual as The initial multipole of a few spheres could not be conclusively deduced as any one particular structure or charge. For such a small system the tachyon mass may be high, leading to a very small though significant exponential attenuation of "lattice energy", which is better described as molecular-like potential energy. The large value for the magnetic charge is not an impediment for this state and there are only a few spheres in total. The multipole and extremely small value of r, provides some qualitative features of a singularity. Such a multipole may act as a seed,

Lattice

Number for replication and expansion, to a crystal. The last sentence is quite speculative and the author regrettably is unable to expand on this.

The most likely candidate for the initial crystal state is the j=26, NaCl structure. A j=13 state is found for a small unfamiliar value of Coulomb charge. However, for reasons discussed in the next four sections, this state does not appear to be the final state. A more detailed discussion of Table 2 is presented in section 8, which includes the interesting series of states for j=1.

Excitations within the Rotational Field

The strong (nuclear) interaction is the order of ħc, which is 137 times greater than that between to two Coulomb charges (e) and 34 times less than that between two magnetic charges (g). We need to examine what type of change within the (N+1)-sphere may provide an interaction of that magnitude. We shall examine a shift in the parameter N and a shift in the parameter L.

The equatorial grand-circle of the N+1 sphere is a N-sphere. The analogy is the equator of radius R of a ball (2sphere) of radius R, has been reduced (to be a 1-sphere) by removing one dimension in the equation of the angularonly-Laplacian. In the ball case (or planet) the dimension removed in the north-south axis. Note that this Laplacian also holds for a thin cylinder if the dimension normal to the circle is "passive" in the sense that all angular momentum is around this axis. Rewriting the equation for angular momentum squared,

〈𝑝 〉 = 1 𝑟 〈𝐿 〉 = ħ 𝑟 𝐿(𝐿 + 𝑁)
For a N+1 sphere. The difference in rotational energy is

〈𝑝 𝑐 〉 / -〈𝑝 𝑐 〉 = ħ𝑐 𝑟 𝐿(𝐿 + 𝑁) -𝐿(𝐿 + 𝑁 -1) = ħ𝑐 𝑟 𝐿(𝐿 + 𝑁) -𝐿(𝐿 + 𝑁) -𝐿)
which, for large L, on expanding the square root for very small (1/L), gives

〈𝑝 𝑐 〉 -〈𝑝 𝑐 〉 ≈ ħ𝑐 2𝑟
If, however L (for the N+1 sphere) is increased from L to L+1 , then 2 (𝐿 -1)(𝐿 + 𝑁 + 1) which similarly decreases as 1/L. Now consider the Coulomb charge qe on this site. If this charge (at only this site) changes for q, then the change in lattice energy for just this site () is given by

𝛿𝜀 = 𝑀 𝛼 2𝑟 𝑞 𝑒 (𝛿𝑞 ±2 )
noting M is negative and the factor of ½ results from the nearest neighbour distance is 2r. Now if q±2 is the same sign as qe then  is negative (as there is a boost in the ionic (or spherical) charge), and if a different sign, then  is positive.

This suggests a small increase (or decrease) in rotational energy at a site corresponds to a small boost (or reduction) in the Coulomb charge at this site. In simple classical terms the increase in speed leads to an increase in charge, which is consistent with Kaluza Klein [START_REF] Th | Zum Unitätsproblem in der Physik Sitzungbar[END_REF][START_REF] Klein | Quanten theorie und fünfdimensionale Relativitätstheorie[END_REF]. Note although L, which effectively describes the number of cycles (of rotational orbit) per second, has decreased from L to L-1, the additional dimensions has effectively increased the path length for each cycle, and a small increase in kinetic energy is realized. This implies of rotational energy required to excite from (N,L) to (L+1,N-2) is through as increase of qe by q-2 and the release of rotational energy from the deexcitation from (L,N) to (L-1,N+1) can decrease qe by q-2. This gives the following expression

𝑁 -1 2𝐿(𝐿 + 𝑁) = -𝑀𝛼𝑞 𝑒 𝛿𝑞 +2 + 𝛿𝑈 -2 𝑁 + 1 (𝐿 -1)(𝐿 + 𝑁 + 1) = +𝑀𝛼𝑞 𝑒 𝛿𝑞 -2 + 𝛿𝑈 +2
Noting M<0 and the residual energy U is expected to be small, otherwise the formation of these excitations would be excessively hot. The U may be partially consumed by forming the apparent mass of the excitation representing a particle. The ratio ofq+2 to q-2 is given by 𝛿𝑞 -2 𝛿𝑞 +2 ≈ -𝑁 + 1 𝑁 -1 which is now assumed to exactly equals. Thus, if the charge (q) has (quark-like) values like +2/3 (and -1/3), then N=3. The value N=3 gives the charge associated with increasing from N=3 to N=3+2, being twice that, and of opposite sign, to the charge associated with reducing from N=3 to N=1.

Note that these values of (N+2) refer to the number of dimensions of the active hyper-(N+1)-sphere, and does not include the inactive dimensions. Inactive dimensions form the hyper-axis, around which the mass rotates. The value of N=3 implies 5 active dimensions, a 4-sphere rotating about an axis. Remember these excitations are limited to a single sphere. The cases of interacting excitations, and a high density of excitations, are discussed in section 7.

Colour Dynamics

This section is a tentative description of the colour charge process in light of the work above. This is speculative work, which may be revised.

Increasing the number of two active dimensions by two, extends the number of active dimensions from 5 to 7. The magnetic and Coulomb compact dimensions are always active. That leaves five other active dimensions. We must have at least seven dimensions, and a minimum of 8 is assumed. These consist of 3 Euclidean, one associated with Coulomb charge, one associated with Magnetic charge and 3 associated with three colour charges (labelled A, B and C). All five dimensions associated with charge are assumed to be compact. The cylinder condition of the Kaluza-Klein model [START_REF] Th | Zum Unitätsproblem in der Physik Sitzungbar[END_REF][START_REF] Klein | Quanten theorie und fünfdimensionale Relativitätstheorie[END_REF] is assumed for compact dimensions.

If all three colour dimensions are active at one time, then the interaction (force) is two dimensional (within 3D Euclidean space). If only two-colour dimensions are active then the interaction can be three dimensional (in 3D Euclidean space). Consider the case of two compact colour dimensions active (A,B) at a particular hypersphere, with all three Euclidian dimensions active. In a brief moment of time, we have three colours (all compact) active during which A is cancelled by anti-A and replaced by C, A unchanged. We now have (B,C). This has been obtained by absorbing a two colour boson consisting of (anti-A,C). This can be also induced by emitting a two-colour boson consisting of (A,anti-C). During this moment of time only two of the Euclidean dimensions are active. A tentative process of six steps is 1. The sphere has two colour dimensions active (A+B), 2. The sphere has three colour dimensions active (A,B,anti-A,C). One Euclidean dimension is inactive.

3. The sphere has two colour dimensions active (B+C), 4. The sphere has three colour dimensions active (B,C,anti-B,A). One Euclidean dimension is inactive. 5. The sphere has two colour dimensions active (A+C), 6. The sphere has three colour dimensions active (C,A,anti-C,B). One Euclidean dimension is inactive. Return to step 1.

Excitations arising from a reduction (by 2) in spatial dimensions from N=3 (a 4-sphere) to N=1 (a 2-sphere) is interesting we now only have 3 active dimensions. These seem to be colour A compact, Coulomb compact, and Magnetic compact. The other four (or more) dimensions must be inactive. So how can we get one compact dimension of colour charge. The only possible answer, assuming the theory is sound, is all the Coulomb charge has vanished to be inactive, and transferred to be a colour charge.

We now have another process of six steps. All steps have no active spatial Euclidean dimensions. The compact magnetic charge dimension is always active.

1. The sphere has one active compact colour (A). Active Coulomb compact dimension. 2. The sphere has two active compact colour dimensions (B=A+anti-A+B). No active compact Coulomb compact dimension. 3. The sphere has one active compact colour (B). Active Coulomb compact dimension. 4. The sphere has two compact colour dimensions active (C=B+anti-B+C). No active Coulomb compact dimension. 5. The sphere has one compact active colour (C). Active Coulomb compact dimension. 6. The sphere has two compact colour dimensions active (A=C+anti-C+A). No active Coulomb compact dimension.

and this switching of the four colour/Coulomb (dimensions) from one colour active (and two inactive) and one Coulomb, to two colour active (and one inactive) with Coulomb compact dimension inactive. This switching is a continuous rotation of the two-dimensional hyper-equator (about an axis of 2 inactive dimensions) within the threesphere (4 dimensions, consisting of one compact Coulomb and three compact Colour charge). When the Coulomb charge is active, the value is q+2=±1/3. For the previous case of seven active dimensions and charge q-2=∓2/3, there is a one-dimensional equator (circle of fixed radius), about an axis of one dimension, that also rotates on the surface of the 3-sphere.

For this process to be valid we need to interaction of the Coulomb charge to be the same order as colour charge which is ħc. Therefore, we need the magnitude of Coulomb charge to be approximately () -1/2 e (where e is the magnitude of the electron charge). Therefore, the relative magnitude Colour charge should be the order of 12e. Again, we propose a K-K process for this charge with magnitude of colour charge proportional to speed of rotating mass in that colour dimension.

The apparent disappearance of Coulomb charge occurs when the two-colour boson is absorbed or emitted. This step is assumed to be a very short fraction of the time.

Pionic Crystal

In this section excitations with pion-like properties are discussed. The term "pion" is used loosely as spinors are not employed, particularly for the two component excitations. The author uses this term associated with spin S=0, mindful that the W and Z particle (Spin=1), may be hidden.

We show in the previous section that q may change to qe+q, where q-2 (N=3 to N=1) is of the opposite sign to qe and q+2 (N=3 to N=5) is the same sign as qe and the magnitude of q+2 is twice that of q-2. Therefore, we may see a "charged pion" form if a positively-charged sphere is excited by q+2 and a negatively-charged sphere is excited by q-2. Or we may see a "neutral pion" if the positive sphere and the negative sphere have the same change (in value for N). These "pions" require the two components to be located with range to interact and bond.

In the limit of a dense pionic crystal, we can consider all spheres undergo these changes. Here all spheres are assumed to have charge qe+q-2, or N=1 (qe-⅓). Another system would be all spheres are assumed to have charge qe+q+2, and N=5 (qe+⅔). Now we examine excitations of a pionic crystal. Assume a crystal where all spheres have Coulomb charge ±qe=±(J+⅓), and N=5, where J is an integer. The change in kinetic energy, for a single hypersphere, is 〈𝑝 , 𝑐 〉 / -〈𝑝 , 𝑐 〉 ≈ + ħ𝑐 𝑟 𝑁 -1 2 𝐿(𝐿 + 𝑁) describing the change from N to N-2. As N=5 then and the new state for this single hypersphere will have N reduced to 3, and Coulomb charge qe will by 2/3, from J+⅓ to J+1.

This reduction (q=⅔) leaves a hole (effective charge -⅔) and "particle" (effective charge (+⅔). Similar changes may occur on for the sphere of opposite charge with q= -⅔. Both holes and particles may hop from sphere to sphere. A particle may transform from a charge of +⅔ to one of -⅓ through the absorption of a boson of Coulomb charge +1 or emission of a particle (excitation that can move) of Coulomb charge -1.

One possible state for the boson is an excitation from N=5, L to N=1, L-4. The relevant release of kinetic energy is summation of the following two equations 2 𝐿(𝐿 + 𝑁) which equates to the Coulomb charge of unity (q=1), since for N=5, the numerator (2N-4) = 6, whereas before (N-5) = 4 equated to ⅔. An excitation with a full unit of Coulomb charge may be related to a lepton or a W particle. Establishing a confident link requires considerable more work.

A resonating pionic crystal is a crystal which has two states, one regular (integer J) and the other pionic (J+⅔ or J+⅓). Such an oscillating state may assist in providing effective Lorentz symmetry as long-range periodicity may be attenuated.

The structure of Euclidean Space

We can summarise the above discussions to three criteria or conditions for a crystalline-like state for threedimensional space. These are Condition 1. must be satisfied. This yields values of qe, j, N and L. Very high accuracy is required, with a value for a to within (at least eight) and preferrably ten significant figures.

The equation

Accruracy of the order of ten significant figures should be expected for the first crystalline phase. A value of N=3 should be expected for the final crystalline phase, unless pionic crystal is found. If the latter occurs, then the appropriate values for q and N should be expected for any state. A minor disagreement may indicate the release of free energy (for a change in crystal phase), or a very large tachyon (imaginary mass mass) for an initial multipole (finite size, not a crystal). The large tachyon mass may attenuate interaction, on induce gravitational repulsion (imaginary mass) for the initial small multipole. The value of qe is the order of 1/ √ 𝛼, which implies L is the order of

-𝑀√𝛼

which gives L the order of 40, as well at qe the order of 12. This estimate is only for the final crystal state and cannot for be applied any transitional states. However, this is completely different to that indicated in earlier work, wrongly estimating L the order of 10,000. These estimates for L and qe should be treated as a means of locating the required order of magnitude and not quantitative. However, the condition 1 (matching the value for ) is strict requiring high accuracy, as well as the value of N=3 and the values for q.

The final step, of this initial investigation of colour dynamics, is to present a possible scenario for the formation of Euclidean 3D space. The results are presented in Table 2. First is step the formation of a multipole, for which a dipole (of two spheres) for N=0 and j=227 is the best fit. This leaves just two active dimensions, assuming one compact dimension for magnetic charge, and one for the dipole axis (quasi-Euclidean). This multipole is the seed which other multipoles will attach and expand to form a three-dimensional crystal. Other potential multipoles are the j=39 within a quadrupole, and the j=66 within an octupole. The value of  is the order of 10 -8 and N=1 for these two, compared to a smaller value less than 10 -9 for the dipole. All mulipoles have no Coulomb charge, as such would increase the value for . 3: Residuals, E and , for the NaCl lattice for several values of L, N, qe and j. terms are calculated for the most accurate published measurement for  [START_REF] Morel | Determination of the Fine Structure Constant with an accuracy of 81 parts per trillion[END_REF]. The terms in the last two columns are the residuals E and .

L

The preferred initial crystal is the NaCl lattice with P-spheres of dyons where the magnetic charge is 26g. This crystal is attracive for two reasons. Excellent agreement for the fine structure is obtained for N=10. Three others sets of Coulomb charge, L and N also provide reasonable values for  are also displayed Table 2.

The value for L for the NaCl is about 10,000 about three orders of magnitude to high than that deduced from Conditions 2 and 3. We require the number of active dimensions to be either N=1 or 3 or 5, so we have q1= -2q2 for two excitations. The preferred lattice is NaCl, j=1, L=37 (and 38), and qe=27, and consistent with Conditions 1 to 3. This lattice is a resonating pionic lattice, N=5, oscillating between (qe=27) and ( qe=27⅓). Note the values of E for these two crystal states have opposite sign and the magnitude agrees to the fifth significant figure for the energy residual E. The magnitude for E is greater for the q=27⅓, while the magnitude of the residual, , is larger for the q=27 state. The measure E is best for separate systems (crystals) while  is best for coexisting systems (crystals). This can be resolved by a resonating state with finite spatial and temporal coherence, allowing for the different states to interact. Now we consider the residuals . These are almost unity. Yet they occur for different values of qe. Something very interesting appears to be going on. The author assumed the wurtzite structure, without a point of inversion, as the strong candidate for basic crystal structure. However, the NaCl lattice appears to the structure. What can a shift in rotational energy of unity times ħ𝑐/𝑟? We initially test 𝐿 (𝐿 + 𝑁 ) = 300 ± 1 finding 43x7=301 and 23x13=299 as the prime factors. A value for N1=36 (L1=7) has an excessive number of dimensiond and dismissed, leaving N1=10 (L1=13) as the only potential candidate. A more complex situation is ½[𝐿 (𝐿 + 𝑁 )+𝐿 (𝐿 + 𝑁 )] = 300 ± 1 which is satisfied by L2=16, N2=3, L3=14, N3=7, giving 299.

We now have a lattice with all sites have the same charge qe=27⅓, half the sites with L=16, N=3; and half with L=14, N =7. Excitations of Coulomb charge 2/3 and ±1, may also occur. The key expression is from Condition 2, 𝛿𝑞 ± = (𝑁 ∓ 1) 6 which which defines the charge but not the mass of the excitated state. Thie mass may vary with the values of N and qe. For example N+1=4 and N-5= 4 show potential 2/3 states. Also qe=27 and qe=27⅓, (N=3 unchanged, N-1=2) show 2 potential 1/3 states. These may indicate the two flavours with low values of mass. The third flavour, with much higher mass, requires a broad search of potential changes in qe and N. One avenue for future work. This slightly imperfect fit of pure crystals is interesting as this will help remove long range order. We do not want all mass to belong to belong to a simple system (built from the well-known fermions and bosons of the Standard Model). Measurements of the Cosmic background have clearly shown that (directly) unobservable dark matter is about three times more massive than observale matter (based on quarks and leptons). The model of the (N+1)-sphere can produce other excitations (or channels) than those discussed in Sections 5 to 7, and the resulting particles may not interact with observable mass beyond gravitational effects.

An example of the process leading to the formation of the deduced Euclidean state (27e & 27⅓e,1g) is: Firstly a dipole of charge j=227, with mi>mr. Secondly formation of the 26g NaCl structure, leading to the forementioned deduce Euclidean state, with mr>>mr. The (j=39) quadrupole is interesting alternative for the initial step (multipole), as the value of j=39 is consistent to those for the lattices. Note the Euclidean state may have at least seven components (listed in Table 3). We require the radius of the (N+1)-sphere to be many orders less than the range of the strong nuclear force so any observed Lorentz symmetry of excited states is insensitive to the periodic symmetry of Euclidean space.

Concluding remarks

This paper presents an interesting model for the initial formation of Euclidean three dimensional space. The dyon charge plays a central role in binding space as an entity, qualitatively similar to ionic-Coulomb charge binds binary salt lattice (crystal). A phase change occurs such the subsequent state for 3D space is still a crystalline NaCl system. This system is a set of at least seven structures, as shown in Table 3, and The dynamics of the (N+1)-sphere provides an interesting basis for the magnitude of Coulomb charge, as multiples of e/3 and a strong hint on flavour.

4 .

 4 Solutions of the equations for L, N, qe and j Determinations can only made within the accuracy of the values of M and . The possible values for L and N are deduced by testing momentum/dimension pairs (L and N being integers) along with magnetic charge and Coulomb charge in the above expression the known value of  and searching for a close match (at least 8 significant figures) for any of the multipole and lattice types listed in

  + 𝑁) -𝐿(𝐿 + 𝑁) + 𝐿 + (𝐿 + 𝑁 + 1))for L being large. An increase in N by one is an effective excitation of ħ𝑐/2𝑟 , while a reduction in N, by one, is a de-excitation of ħ𝑐/2𝑟. An increase in the value of L is an excitation, requiring an energy of ħ𝑐/𝐿𝑟 , and a decrease of L is de-excitation, releasing energy ħ𝑐/𝑟. Together, a decrease of N by 2, and an increase of L by 1, gives 𝐿 + 𝑁) which describes the energy required to induce this excitation. This value decreases like 1/L. The reverse is 𝐿 + 𝑁) which describes a release of energy. By replacing N with N+2, and L with L-1, gives ∆ , = 〈𝑝 , 𝑐 〉 -〈𝑝 , 𝑐 〉 / ≈ + ħ𝑐 𝑟 𝑁 + 1

Condition 2 .

 2 We have 𝛿𝑞 = ±2/3, and 𝛿𝑞 ∓ 1/3, and N=3 is deduced from the equations Note, this relationship is approximate, and depends on the magnitude of U. If the term (U) is much smaller than the two other terms, then the above expression increases in accuracy.

  /m 2 gives

	(ħ𝑐) 𝑟 𝐿(𝐿 + 𝑁)	(𝑚 -𝑚 ) -2𝑖𝑚 𝑚 (𝑚 + 𝑚 )	+ 𝑐 𝑟
		=	(𝐾𝑀𝑞 ) 16	(𝑚 -𝑚 ) -2𝑖𝑚 𝑚 (𝑚 + 𝑚 )	𝑟 -	𝐾𝑞 𝑟 𝑀𝑉 , 2𝑚	+	𝑉 , 𝑟 𝑚	(𝑚 -𝑚 ) + 2𝑖𝑚 𝑚
	3.3.1 Imaginary terms
	Grouping imaginary terms gives
				(𝐾𝑀𝑞 ) 16(ħ𝑐)	-𝐿(𝐿 + 𝑁) =	𝑉 , 𝑟	1 ħ𝑐	𝑚 𝑚	+ 1
	which are rewritten as
				(ħ𝑐) 𝐿(𝐿 + 𝑁) =	(𝐾𝑀𝑞 ) 16	-	𝑉 , 𝑟	𝑚 𝑚	+ 1
	noting that if the second term on the right hand side is much smaller than the first term, then
					ħ𝑐 ≈		(-𝑀)	(𝐾𝑞 )
						4 𝐿(𝐿 + 𝑁)
			(𝐾𝑀𝑞 ) 16(ħ𝑐)	-𝐿(𝐿 + 𝑁) = 𝜀 =	𝑉 , ħ𝑐	𝑚 𝑚	+ 1 =	𝐺𝑚 ħ𝑐	𝑚 𝑚	+ 1

as M is negative. Note for the one-dimensional circle (1-torus, or linear grid), P=1 and

  Noting as ro>>r, suggessting a substantial reduction in the magnitude of mi. The 3D torus, or cubic lattice, gives

		𝜀 =	4𝜋 𝐺𝑚 𝑟𝑐	𝑚 𝑚	+ 1 =	4𝜋 𝐺𝑚 𝑚 𝑟𝑐	𝑚 𝑚	+ 1 = 2𝜋	𝑟 𝑟	𝐺𝑚 ħ𝑐	𝑚 𝑚	+ 1
		𝜀 =	4𝜋 ħ𝐺 𝑟 𝑐	𝑚 𝑚	+ 1 =	8𝜋 ħ𝐺𝑚 𝑚 𝑟 𝑐	𝑚 𝑚	+ 1 = 2𝜋	𝑟 𝑟	𝐺𝑚 ħ𝑐	𝑚 𝑚	+ 1
	as ro=(2ħ/mic), also can use miro=(2ħ/c),
	3.3.2 Real terms
	Grouping the real terms gives
	(ħ𝑐) 𝑟 𝐿(𝐿 + 𝑁)	(𝑚 -𝑚 ) (𝑚 + 𝑚 )	+ 𝑐 𝑟 =	(𝐾𝑀𝑞 ) 16	(𝑚 -𝑚 ) (𝑚 + 𝑚 )	𝑟 -	𝐾𝑞 𝑟 𝑀𝑉 , 2𝑚	+	𝑉 , 𝑟 𝑚	(𝑚 -𝑚 )
	which, on using the results of the imaginary terms, is simplified to a quadratic in terms of VG/(𝑟 mic 2 )
			𝑟 = -	𝐾𝑞 𝑀 2(𝑚 𝑐 )	𝑉 , 𝑟 (𝑚 𝑐 )	+ 2	𝑚 𝑚	-1	𝑉 , 𝑟 (𝑚 𝑐 )

Table 1 :

 1 which should be close to zero for three dimensions if the exact values of M and  are known and used. Another related quantity (E) is an energy residual given by Values of Madelung Constant M for spatial geometries and number of non-compactified dimensions.Values for the Madelung constants are displayed in Table1above. We have the following key expression

	𝛿 = 8𝜋	ħ 𝑐 𝐾𝑞 (-𝑀)	𝑚 𝑚	+ 1	𝑟 𝑟	= 𝐿(𝐿 + 𝑁) -	(𝐾𝑀𝑞 ) 16(ħ𝑐)
		𝛿 = 𝐿(𝐿 + 𝑁) -	(𝐾𝑀𝑞 ) 16(ħ𝑐)

Table 2 :

 2 table 1[START_REF] Tavernier | Clifford boundary conditions: a simple direct-sum evaluation of Madelung constants[END_REF][START_REF] Zucker | Madelung constants and lattice sums for hexagonal crystals[END_REF]. Relevant examples are shown in Table2. Note that there is a preference for integer values for j. Residuals, , for values of L, N, n and j. The terms are calculated for the most accurate published measurement for [START_REF] Morel | Determination of the Fine Structure Constant with an accuracy of 81 parts per trillion[END_REF]. The terms outside brackets are the residuals

	L	N Lattice type	j	qe	L(L+N)	 x  
	441113 0	dipole	227	0	194,774,816,889 -170 (8.7)
	16482	1	quadrupole	39	0	283,669,806 -5.2809(183.7)
	10143	10 NaCl	26	1/16	102,373,899 -0.0062 (-0.6069)
	10144	8	NaCl	26	6/16	102,373,908 0.1876 (18.14)
	10145	6	NaCl	26	8/16	102,373,915 0.1294 (12.36)
	10146	0	NaCl	26	10/16	102,373,918 0.0569 (5.56)
	2529	1	NaCl	13	147/512	6,398,370 0.00089 (1.39)
	2529	1	NaCl	13	0	6,398,370 1.3286 (2,076)
	2528	3	NaCl	13	0	6,398,368 -0.6714 (1,049)


B Satyanarayana

B Thierry

D Lo Seen

A V Raman

G Muthusankar

REMOTE SENSING IN MANGROVE RESEARCH -RELATIONSHIP BETWEEN VEGETATION INDICES AND DENDROMETRIC PARAMETERS: A CASE FOR CORINGA, EAST COAST OF INDIA

Keywords: Mangroves, Remote Sensing, PCQM, Vegetation Index

The mangrove forest of the Godavari estuary, Andhra Pradesh, represents the second largest area of such vegetation formations along the East Coast of India, next to the Sunderbans (West-Bengal). Although declared as Wildlife Sanctuary since 1972, this rich but fragile ecosystem has undergone serious alterations largely induced by human activities. Continuous efficient retrieval of reliable information from the mangroves is therefore necessary for conservation purposes. Satellite remote sensing is a useful source of information as it provides timely and complete coverage of the study area, complementing field surveys of higher information content but which are more difficult to carry out, especially in mangroves. The purpose of the present study is 1) to map the mangrove formations and its surroundings based on a supervised classification of remote-sensing data and 2) to analyse the potential relationships between mangrove dendrometric parameters and spectral indices extracted from satellite data. The supervised classification was carried out with an IRS-1C LISS3 image of March 1999 and was trained from ground truth data and field knowledge. Among the resulting 14 classes, 3 correspond to different mangrove signatures. The ground truth includes 128 sampling locations for which mangrove vegetation parameters like basal area and tree density have been estimated using the Point Centred Quarter Method (PCQM) on transect lines of at least 100 m. In a second stage, Vegetation Indices (VI) have been calculated at locations for which mangrove parameters were obtained from field surveys. Various statistical tools, among which scatter-plots and analyses of variance (ANOVA), have been used in order to explore the relationships that may exist between VI and mangrove parameters. The first results show that a relationship exists between VI and basal area whereas this is not the case with density. Furthermore, when spectral indices and mangrove parameters are considered altogether, it appears that only two classes of mangrove can be discriminated.

INTRODUCTION

Mangroves represent a specific ecosystem found in the intertidal zone along tropical and subtropical coastlines, and are often located near estuaries and deltas [START_REF] Spalding | World Mangrove Atlas[END_REF]. Being highly productive ecosystems and harbouring a large diversity of species adapted to these particular habitats, they are considered of utmost ecological importance. Moreover, they provide a number of direct and indirect services, ranging from protection against coastal erosion [START_REF] Pearce | An unnatural disaster -Clearing India's mangrove forests has left the coast defenceless[END_REF] to the multiple forest products usage by local population [START_REF] Blasco | The Mangroves of India[END_REF]. For the past decades however, the situation of mangrove forests has been continuously deteriorating due to an increasing human pressure resulting in conversion to agricultural lands, renovation of brackish water fisheries, prawn and shrimp farms, salt pans, urban and industrial pollution, etc. [START_REF] Clough | Mangrove Ecosystems in Australia: Structure, Function and Management[END_REF].

The mangrove forest of the Godavari estuary, Andhra Pradesh, is the second largest area of such vegetation formations along the East Coast of India, next to the Sunderbans (West-Bengal) and counts about fifteen mangrove species, among which Avicennia marina, A. officinalis and Excoecaria agallocha are the most dominant ones. Although declared as Wildlife Sanctuary since 1972, this rich but fragile ecosystem has undergone serious alterations largely induced by human activities. Continuous efficient retrieval of reliable information from the mangroves is therefore necessary for conservation purposes.

Satellite remote sensing is a useful source of information as it provides timely and complete coverage of the study area, complementing field surveys of higher information content but which are more difficult to carry out, especially in mangroves. For these reasons, studies have been carried out on mangrove ecosystems using aerial photography (e.g. [START_REF] Dahdouh-Guebas | Four decade vegetation dynamics in Sri Lankan mangroves as detected from sequential aerial photography: a case study in Galle[END_REF], optical (e.g. [START_REF] Rasolofoharinoro | A remote sensing based methodology for mangrove studies in Madagascar[END_REF], and radar (e.g. [START_REF] Mougin | Multifrequency and multipolarization radar backscattering from mangrove forests[END_REF] remote sensing data, or a combination of them (e.g. Pasqualini et al., 1999). However, the objectives of the studies differ according to what can be expected from the different types of remote sensing data. For example, mapping mangroves at the species level can be attempted with high-resolution aerial photography, whereas mapping the landscape level environmental indicators of a coastal area can generally be carried out using optical satellite images from sensors like Landsat TM, SPOT HRV or IRS LISS [START_REF] Klemas | Remote Sensing of landscape-level coastal environmental indicators[END_REF][START_REF] Ramachandran | Application of Remote Sensing and GIS to coastal wetland ecology of Tamil Nadu and Andaman and Nicobar group of islands with special reference to mangroves[END_REF]. For the estimation of mangrove forest parameters like basal area or biomass, radar remote sensing seems most promising [START_REF] Mougin | Multifrequency and multipolarization radar backscattering from mangrove forests[END_REF][START_REF] Proisy | Interpretation of polarimetric radar signatures of mangrove forests[END_REF], although appropriate configurations of frequency, polarisation and spatial resolution are currently not available on orbiting radar satellites.

In the present study, we explore the possibility of using spectral indices derived from optical satellite images to give quantitative estimates of mangrove vegetation parameters. The study area, the Coringa Forest, is one of the most closely followed mangroves in India for the last decade. Significant ground data of the area have been acquired during numerous measurement campaigns in the mangrove forests during the last five years. In a first step, the mangrove formations and its surroundings are mapped, based on a supervised classification of an IRS-1C image of March 1999. In a second step, the potential relationships between mangrove dendrometric parameters measured on the ground and spectral indices extracted from the satellite data are analysed and interpreted.

MATERIALS AND METHODS

Study area

The Godavari is the second longest river of all the Indian sub-continent. It divides into Gautami and Vasista just after the Dowlaiswaram Dam about 60 km before reaching the Bay of Bengal. The mangroves studied are located around the Gautami-Godavari estuary. The study area extends from around 82°05'E to 82°25'E and 16°30'N to 17°05'N and includes Kakinada city, Kakinada Bay, the Coringa Wildlife Sanctuary where the most important stretch of mangrove is found, and also the mangrove forest situated south of Gautami-Godavari River. Several different landscapes compose the area (Figure 1), with paddy fields and coconut tree plantations in the west and south, mangrove forests, aquaculture ponds for shrimp farming spreading into mangrove forests, saltpans, casuarinas plantations along the beach and on Hope Island, villages and urban areas (Kakinada, Yanam).

Ground data collection

The ground data used in the present study were acquired during the period 1998-2000, and consist mainly of mangrove vegetation identification, counting and dendrometry. A regular sampling grid of 1-minute latitude and longitude spacing is first used. Additional sampling points have also been taken, mainly in the mangrove area so that, out of a total of 128 sample plots visited, 83 pertained to the mangrove forest (Figure 1). Each sample plot was accessed with the help of a GPS receiver (model Garmin 45) with an estimated accuracy of 100 m due to the Selective Availability (SA) introduced by the US Department of Defence at that time, i.e. before May 2000. When any given grid node could not be reached, the sample plot was installed at the nearest accessible place and the surrounding land-use recorded.

For the mangrove sampling plots, data acquisition was done using the PCQ-Method (Point Centred Quarter Method) [START_REF] Cintrón | Methods for studying mangrove structure[END_REF]. This allows for the measurement of density, basal area, mean diameter and relative composition of forest stands. For each mangrove plot, a transect line of at least 100 m (depending on accessibility) was laid out westwards. Every ten meters along the transect, four quarters were established by drawing a line perpendicular to the transect line. Then, in each quarter, the tree nearest to the node was measured (Figure 2). Following the recommendations of [START_REF] Cintrón | Methods for studying mangrove structure[END_REF], diameter at breast height (dbh) was obtained by measuring girth at 1.3 m from the ground in the case of erect tall trees (Avicennia marina, A. officinalis, A. alba, Bruguiera gymnorrhiza, Excoecaria agallocha, Sonneratia apetala and Xylocarpus mekongensis) and above the highest established prop roots for Rhizophora trees. In the case of individuals less than 3 m high (mainly Aegiceras corniculatum, Bruguiera cylindrica, Ceriops decandra, Lumnitzera racemosa and some A. marina), the girth was measured below the lowest branch point. All the measurements made did not require any sophisticated equipment. For that reason, data like Leaf Area Index (LAI), which could have been interesting for this remote sensing study, were not acquired. The data collected were entered in a spatial database for subsequent spatial queries and analysis.

Satellite data analysis

The multispectral image used in this study is an IRS-1C LISS3 image of March, 8 th 1999 with 4 spectral bands (green, red, near infrared (NIR) and short-wave infrared (SWIR)). The spatial resolution (expressed as pixel size) is 23.5 m for the three visible/NIR bands and 70.5 m for the SWIR band. The image has been geocoded in UTM projection based on ground control points obtained by post-SA GPS measurements. The RMS error of the transformation was less than 15 m. One of the goals of the remote-sensing data analysis was to produce a land-use map of the mangrove forest and its surroundings. This was done in a two-step process. First, a maximum likelihood supervised classification was carried out using training areas chosen according to extensive field knowledge but without any specific reference to the grid sample points. Afterwards, the raw result of the supervised classification was checked during visual interpretation of the satellite image and field visits. Small polygons that were obviously wrong (e.g. sediment plumes into the bay were classified as fallow land) have been recoded so as to match the operator's field knowledge (Figure 3).

The second part of the study concerned the analysis of potential relationships between vegetation indices and mangrove dendrometric parameters (namely, density and basal area). Normalised Difference Vegetation Index (NDVI) was calculated from the image as the band ratio (NIR -Red)/(NIR + Red). The transect locations were then entered in the spatial database as rectangles of roughly 150 m x 75 m, which correspond to 18 pixels. This size represents a compromise between the actual dimensions of the field transects and a pixel set large enough to minimise statistical bias due to the spatial variation of pixel value. In the analysis, only sample plots (83) in mangroves were considered. Each transect location on the image was reviewed to account for the 100 m position uncertainty given by the GPS. This prevented overlapping with neighbouring categories of land-use like beach, river and barren land. The mean value of the area thus covered by one transect rectangle was then extracted from the NDVI image and the land-use map. This value was then stored for each mangrove sample plot in the spatial database along with the other available information.

1 6 ° 3 8 ' 16°38' 1 6 ° 4 2 ' 16°42' 1 6 ° 4 6 ' 16°46' 1 6 ° 5 0 ' 16°50' 1 6 ° 5 4 ' 16°54' 1 6 ° 5 8 ' 16°58' 1 7 ° 2 ' 17°2' 8 2 ° 1 0 ' 8 2 ° 1 0 ' 8 2 ° 1 4 ' 8 2 ° 1 4 ' 8 2 ° 1 8 ' 8 2 ° 1 8 ' 8 2 ° 2 2 ' 8 2 ° 2 2 '

Supervised classification

The land-use map derived from the satellite image is presented in Figure 3. The classification led to 14 classes, among which three were for mangrove forest, i.e. dense, medium dense and less dense mangrove. In order to evaluate the accuracy of the land-use map, a confusion matrix was produced [START_REF] Congalton | A Review of Assessing the Accuracy of Classifications of Remotely Sensed Data[END_REF] using the 128 ground truth locations. As the classes used for ground truth data were different from those of the land-use map, a

Study area

non-bijective correspondence had to be set between them. This took the form of a lookup table as presented in Table 1. After overlaying the ground truth locations on the land-use map, the land-use classes corresponding to each of them were noted. Given the relatively small number of locations considered, each individual location was checked both on the image and land-use map to avoid residual misplacement due to SA uncertainty. Plots that fell on or very near boundaries between aquaculture ponds and paddy fields were assigned to the mixed class. The results are given in the confusion matrix shown in Table 2.

Table 1: Lookup table between land-use map classes and ground truth classes Table 2: Confusion matrix of supervised classification and ground truth

The reading of the confusion matrix draws some comments. First of all, it can be noted that mangrove has been very well classified, especially since the three mangrove classes have been merged into one. This is not surprising as it is displayed with a characteristic spectral signature on the satellite image as shown in false colour composite (FCC) in Figure 1. The two points misclassified are found on barren areas within mangroves and are anyhow close to mangrove vegetation. The confusion between Agriculture and Barren land on one hand and Aquaculture (ground truth) and Barren land (supervised classification) on the other hand can be explained by the seasonal cycles of the paddies and ponds. Periodically, aquaculture ponds need to be emptied for maintenance and cleaning (at least once a year) and during that period, the dried ponds have the same spectral signature as barren land. As the image was taken on March 8 th , 1999 and the fieldwork had been carried out at different periods that spans over 3 years, it is most likely that a number of ponds were empty during the satellite overpass. A similar explanation is also valid for paddy fields: after harvest (which happens once or twice a year), the paddy fields are either left as fallow lands or used for growing leguminous plants (e.g. grams). These sparsely vegetated areas tend to have a spectral behaviour very close to that of barren lands, hence the confusion that arises from the time difference between the image acquisition date and the field visits.

Relationship between NDVI and dendrometric parameters

Here, we try to verify whether the three different mangrove classes obtained from the satellite image are substantiated by mangrove vegetation field measurements. In other terms, potential relationships between NDVI and dendrometric factors (viz. density and basal area) of mangrove forest are investigated. This has been done through qualitative analysis as well as quantitative -statistical -methods.

An efficient statistical method to analyse relationships between a qualitative factor (e.g., mangrove classes) and a quantitative factor (e.g., basal area or density) is analysis of variance (ANOVA). This method investigates, in a set of values organised in several groups, the proportion of variance that can be explained by within-group variability (called Mean Square Error or MS err ) and inter-group variability (called Mean Square Effect or MS eff ). The smaller MS err and higher MS eff , the stronger the relationship between the set of values and the groups. Yet, as shown in Figure 4, mean values for basal area of Less dense mangrove and Medium dense mangrove are not significantly different (respectively, 1.034m 2 /0.1ha and 1.031m 2 /0.1ha) to be considered as two groups with respect to basal area. The same observation also holds for density. Consequently, Less dense mangrove and Medium dense mangrove classes have been merged into one class called Less/Medium dense mangrove. With these two groups (i.e., Less/Medium dense mangrove and Dense mangrove), ANOVA gives MS eff = 97.4 and MS err = 3.3 for basal area on one hand and MS eff = 444193.8 and MS err = 68895.1 for density on the other hand. Using the F test, which measures the significance of the ratio of the two variances, we obtain F = 29.5 for basal area and F = 6.4 for density. Therefore the two mangrove classes can be considered highly related to the basal area and much less pertinent relative to density. These results are clearly illustrated on Figure 5 and Figure 6. The two scatter plots show the relationships between NDVI and basal area (Figure 5) and between NDVI and density (Figure 6) for the three classes, viz. Less, Medium dense and Dense mangrove. In both figures, Less, Medium dense and Dense mangroves are displayed in correct order, that is, with increasing mean NDVI from Less dense to Dense mangrove. This is as expected because both NDVI and the classification were derived from the same data source (satellite image). When NDVI is plotted against basal area, two zones can easily be identified along the main diagonal, one in the lower left quarter, and the other in the upper right quarter of the graph, which indicates that a relationship exists between basal area and NDVI. However, when NDVI is plotted against density, no relationship is found. This is not quite surprising because density expressed as number of individuals per unit area is not a good indicator of the 'amount' of vegetation as seen in the satellite image, as the sizes of the individuals can be very different (height up to 10 m, dbh up to 60 cm). In comparison, basal area is shown to be a better indicator of vegetation 'amount' such that Less/Medium dense mangrove may stand for lower basal area and Dense mangrove for higher basal area. These preliminary results show that at least a broad estimate of the basal area of mangrove forest can be obtained from optical satellite imagery. In fact, the relationship is more between classes of NDVI and basal area than between NDVI and basal area directly. This relationship is therefore considered not significant enough to allow basal area mapping of the mangrove forest.

11 2 - 1 - - - 14 79% Barren land - 5 - - - - 2 7 71% Mangrove - 2 81 - - - - 83 98% Mixed - - - 3 1 - - 4 75% Plantation - - - - - - - 0 N/A Settlement - - - - - 1 - 1 100%

CONCLUSION

A land-use map of the Godavari estuary area was made from supervised classification of an IRS-1C LISS 3 satellite image of March 8, 1999. A difficulty faced while carrying out this classification was to correctly identify some aquaculture ponds and agricultural fields while they were seen as barren lands on the day of the satellite overpass. This type of errors can however be reduced/eliminated using multi-date classification of images taken at different seasons. The mangrove areas have been presented in three classes in the land-use map corresponding to Less dense, Medium dense and Dense mangroves, according to differences in spectral characteristics revealed during the classification procedure. When compared with ground data, these differences were consistent with differences in basal area only after merging Less dense and Medium dense classes, as the basal area measurements for these two classes were not significantly different. In that way, Less/Medium dense mangroves were found to correspond to mangroves with lower basal area, and Dense mangroves, to those with higher basal area. However, no such relationship was found with measured 'density'. This observation can be understood by the fact that the term 'density' has different meanings in the classification and in ground measurements. In the classification, density stands for the 'amount' of vegetation seen by the sensor or by a person walking in the mangroves, whereas the measurements are expressed in number of individual trees per unit surface area. These mangroves being neither mono-specific nor even-aged, it is normal that a same number of trees will correspond to different 'amount' of vegetation. In that sense, basal area has a definition closer to the density used in the classification, which explains the relationship between density classes and classes of basal area. The direct relationship between spectral indices and basal area is however not considered significant enough to allow basal area mapping. A possible explanation is that spectral indices integrate variabilities in optical characteristics, amount and spatial organisation of leaves due to species and edaphic conditions that are only partially expressed in basal area. Conversely, a similar study aiming at species identification would be hampered by variabilities in vegetation densities that are not related to species composition. Nevertheless, recent and future sensors with increased spatial and spectral resolutions are bringing new hope to obtain timely and precise information necessary for managing mangrove ecosystem conservation.

Figure

  Figure 1: IRS-1C image of the study area, the 128 sample plots are also represented

FigureFigure 6 :

 6 Figure 3: Land-use map

ACKNOWLEDGMENTS

The authors wish to thank Dr. L.P. Jayatissa from the University of Ruhuna, Sri Lanka, for helpful discussions and suggestions. Research during 1998-'99 was carried out with funds provided by the Department of Ocean Development, New Delhi and later by the European Commission under Contract No. ERB IC18-CT98-0295. The authors are grateful to Dr. B.R. Subramanian, Director, DOD for encouragement. Facilities at the IFP, Pondicherry and Andhra University were utilised and we thank the authorities concerned.


Valentin Lachand
email: valentin.lachand@liris.cnrs.fr

Aurélien Tabard
email: aurelien.tabard@liris.cnrs.fr

Christine Michel
email: christine.michel@liris.cnrs.fr

Toccata: A Multi-Device System for Activity Scripting and Classroom Orchestration

We present Toccata, a system supporting the management of rich multi-device pedagogical activities. Activities designed with Toccata are reusable, shareable and adaptable to the situation. Teachers face numerous challenges in designing and scripting pedagogical activities that incorporate rich media and applications, combine devices, group formations, and spaces. They also face challenges in orchestrating these activities in class, especially to guiding learners, following their progress, and maintaining a coherent learning experience. Our demonstration will showcase how Toccata answers these challenges by supporting individual or collaborative activities, in class or outdoors, under diverse technical conditions, e.g., offline on mobile devices, or fully connected with Desktop computers.

Introduction

Toccata is an activity-centric system supporting orchestration of pedagogical activities for classrooms. Orchestration aims at helping teachers to create pedagogical scripts, adapt and execute them in a given context [START_REF] Dillenbourg | Design for classroom orchestration[END_REF]. Based on interviews with teachers [START_REF] Jalal | Design, Adjust and Reuse --How Teachers Script Pedagogical Activities[END_REF], we identified a set of recurring challenges for creating and conducting digital pedagogical activities in today's classrooms, such as resilience to networking problems, or support for a wide variety of devices. We developed Toccata to facilitate the management of rich multi-device pedagogical activities. In Toccata, activities are reusable, shareable and adaptable to the situation. Toccata supports tight or loose activity scripting. It lets teachers conduct digital activities in class and in more open environments. Toccata also lets teachers modify and adapt unfolding activities according to the situation in the classroom.

Teachers face numerous challenges in integrating digital tools into smooth and coherent teaching activities. Rich pedagogical activities are often fragmented in time, split into multiple sub-activities, built upon multiple media and applications, and may unfold in various locations. They can be initiated in a specific context (technical, physical or social) and continued in another, following a plan more or less strictly defined by teachers.

For example, in a vocational school, a horticulture lesson can start in class with a lecture leveraging an interactive whiteboard, continue in the school's greenhouse with individual work on tablets to inventory flowers and, finish back in the classroom with a synthesis in groups shared with others. Follow-up activities and homework can finally be conducted outside of the school on personal devices.

Combining various activities, devices, group formations, and spaces brings new pedagogical opportunities such as the scenario previously described. But organizing such scenarios is complex as teachers must be able to guide learners, and follow their progress, while maintaining a coherent learning environment. The lack of tools to support these practices makes it difficult for teachers to put rich pedagogical activities into place. And when such activities do happen, infrastructural problems (network, device set-up, content distribution) make it even more challenging.

We designed Toccata to work in schools with various technical set-ups and policies. Teachers can run activities in their classroom, on fixed computers with a reliable network, but also across several rooms or with multiple devices, even if there is no insurance of a reliable network. Toccata also supports disconnected contexts, such as in sports class or activities unfolding outside of school. We have tested Toccata in three different middle schools with highly varied situations: in classrooms over multiple sessions, in activities mixing digital and physical resources, in nomadic activities.

Our demonstration will showcase the versatility of Toccata. We will present activities created by teachers and run with Toccata such as: 1) a fact checking activity, in which students have to verify and correct wiki articles over multiple sessions; 2) a sales management activity which involves roaming, i.e. work on tablets and on a large display in the classroom, but also documentation in a greenhouse; and 3) a collaborative activity involving paper and tablets, in order to learn Agile project management. We will also discuss the underlying architecture allowing our system to work in schools with diverse technical and policy constraints, and its theoretical grounding

Toccata

Toccata is a Web-based application enabling teachers to create digital pedagogical activities and conduct them with a class. As a Web-based application, our system works with any kind of devices containing a web browser (computer, tablet, phone, video-projector linked to a computer). Toccata is developed to run in online or offline mode, with an optimal mode and two degraded modes according to network reliability.

Activity Scripting

Toccata builds upon an Activity Based [START_REF] Bardram | Activity-based computing-lessons learned and open issues[END_REF] model to represent pedagogical activities. When teachers create an activity, they can define the following components, before enactment or live:

1. A set of Instructions for students to guide them in conducting the activity. 2. A list of Sub-activities (steps), created by teachers to divide the activity according to pedagogical needs. 3. A set of Resources can be associated with activity or steps and are typically all kind of documents openable with a web browser (pdf, video, audio, website, etc.). These resources are read-only in our system. 4. A set of Applications are a set of tools allowing teachers and students to run operations on resources, such as text editors; and to control the activity flow, such as timers. 5. A list of Participants involved in the activity. 6. Notes that teachers can attach to an activity or step, to jot down things to remember or document of the class unfolded, to improve future iterations of the activity.

Activity Orchestration

During the enactment phase, the teacher and the students have a similar interface. However, the teacher can take notes on the activity, and has more options and actions available on the applications. S/he can edit during class each component of the activity. This run-time edition of the activity can help the teacher in the adaptation of the activity for students. For example, if a student is blocked on a particular exercise, the teacher can refine instructions and give new resources to help the student, or s/he can change the order of steps, add new exercises for specific group, change the visibility of steps for several groups.

Scenario

We illustrate the use of Toccata with a scenario developed in collaboration with a teacher from a vocational middle school. The activity was run in the school over a period of 90 minutes, with two groups of learners to test Toccata in-situ.

Thomas is an economics teacher in a vocational middle school. At the end of the year, he decides to create an activity to review the topics covered during the year. He creates the activity with Toccata and divides it in three steps:

1. Students watch a video of a sale situation, analyze it, and answer questions about concepts from previous course chapters presented in the video in a text editor. They can use the pdf viewer to read the previous lessons related to the video. 2. Students calculate taxes and prices using a collaborative spreadsheet. 3. Students prepare the plant catalog of the school. They visit the greenhouses of the school, take pictures of plants with the tablet and store them in their space in order to build the catalog. At the end of the first two steps, Thomas includes a correction: a student will do the exercise on the teacher environment in front of the class and project it with the classroom's video-projector. At the end of the third step, Thomas includes a class discussion: each group will present the pictures taken in the greenhouse by using the video-projector and discuss with the class how they could be used to create the sale catalog.

In the first phase, students analyzed a sales situation using external applications within Toccata. Due to an unreliable school network, external Web applications such as the collaborative text editor did not work perfectly, but resources hosted on Toccata could be properly accessed. Although the internet connection was fluctuating, students did not encounter major problems, and managed to move from one step to the next. During the second step, due to the use of an external application for the spreadsheet, changes made during the correction on the teachers' computer connected to the video-projector were not synchronized with students' activity, and students had to manually update the activity on their tablet. In the third step, students moved to a greenhouse, with no WiFi coverage. This step worked smoothly and students could grab their tablet and continue their activity as expected. In the greenhouse, they moved freely and took pictures of plants they later added to a sales catalog. When they came back to the classroom, the activity was updated on the global server and the teacher could access the pictures taken in the greenhouse.

Implementation

Toccata architecture is built on three layers. The first layer consists of a main Web server and Web applications. The second layer is a local server running inside classrooms. The last layer is composed of client devices running Toccata. The second layer is not mandatory, and devices running Toccata can directly communicate with the remote Web server. When the third layer is not available, the devices synchronize with each other via a local server if they are connected to one. Otherwise they run independently.

Toccata is a Progressive Web Application built with Angular, with duplicated layers, and extra synchronization mechanisms. The server delivers a single page application running in the browser. It is hosted on Firebase. Activities are stored on a CouchDB database. External applications are iframes opened inside Toccata and local applications are Angular components.

The local server acts as a WiFi hotspot, it can either connect to the Ethernet network of the school, or connect to a tethering smartphone, or run without any Internet connection. The server runs on Node.js and only delivers a simple Single Page Application with very little logic. The activities are stored in a CouchDB database that syncs with the remote one.

Each device runs Toccata and a PouchDB instance. PouchDB allows synchronization between multiple instances of CouchDB servers. As a Progressive Web Application, is offers some native-like feature, like home icons on mobile OS, and strong caching mechanisms for the data, but also for the application shell: the webpage can load even when the device is totally offline, and it will synchronize back to the server when it becomes available.

Toccata is open-source and available on GitLab (https://gitlab.com/lachand/Toccata). Since Toccata is built on Angular with reusable web components, everyone can reuse components of Toccata in its own project and we think that this may lead to a better integration and assimilation of research projects in commercial and deployed in school systems. In addition, we create a demonstration page where people can use Toccata with a student or teacher account in order to enable teachers, researchers or system editors to test our system and concepts beneath Toccata. This demonstration is available at the following page: https://demo.toccata.education

Future work

In future work, we will study the use of Toccata in different contexts, such as a collaborative activity with one device per student instead of one device per groups. We will also work on a classification of tasks and new interaction to support teachers when they had to manage an activity with several devices (attention management, support to students, etc.). Another interesting point fur future work is to pass from a note-taking process (what is actually done by teachers) to a reflexive process to design and re-design pedagogical activities.

Conclusion

We presented Toccata, a system allowing teachers to prepare multi-device activities before class. Toccata enables teachers to edit their activity at it occurs, in order to adapt it to students' progression or to unexpected events occurring in class. Toccata has been tested in different schools and pedagogical context, such as a course on several sessions, a course mixing paper and digital activities, and a course running in class and outside class, with no available network. The demo of Toccata offers participants the ability to edit and run existing activities created by real teachers, and to create their own scripts. Participants will be able to run their scripts on tablets during the demonstration.

Fig. 1 :

 1 Fig. 1: A preview of an activity in Toccata, and components of the activity

Fig. 2 :

 2 Fig. 2: (a) Tutor view of Toccata when looking at different group progression; (b) Student view of Toccata during an activity with a project management board and timer loaded