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Archimedean solid

In geometry, an Archimedean solid is one of 13 convex polyhedra whose faces are regular polygons and whose vertices are all symmetric to each other. They were first enumerated by Archimedes. They belong to the class of convex uniform polyhedra, the convex polyhedra with regular faces and symmetric vertices, which is divided into the Archimedean solids, the five Platonic solids (each with only one type of polygon face), and the two infinite families of prisms and antiprisms. The pseudorhombicuboctahedron is an extra polyhedron with regular faces and congruent vertices, but it is not generally counted as an Archimedean solid because it is not vertex-transitive. An even larger class than the convex uniform polyhedra is the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.

In these polyhedra, the vertices are identical, in the sense that a global isometry of the entire solid takes any one vertex to any other. observed that a 14th polyhedron, the elongated square gyrobicupola (or pseudo-rhombicuboctahedron), meets a weaker definition of an Archimedean solid, in which "identical vertices" means merely that the parts of the polyhedron near any two vertices look the same (they have the same shapes of faces meeting around each vertex in the same order and forming the same angles). Grünbaum pointed out a frequent error in which authors define Archimedean solids using some form of this local definition but omit the 14th polyhedron. If only 13 polyhedra are to be listed, the definition must use global symmetries of the polyhedron rather than local neighborhoods.

Prisms and antiprisms, whose symmetry groups are the dihedral groups, are generally not considered to be Archimedean solids, even though their faces are regular polygons and their symmetry groups act transitively on their vertices. Excluding these two infinite families, there are 13 Archimedean solids. All the Archimedean solids (but not the elongated square gyrobicupola) can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry.

Origin of name
The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra. During the Renaissance, artists and mathematicians valued pure forms with high symmetry, and by around 1620 Johannes Kepler had completed the rediscovery of the 13 polyhedra, as well as defining the prisms, antiprisms, and the non-convex solids known as Kepler-Poinsot polyhedra. (See for more information about the rediscovery of the Archimedean solids during the renaissance.)

Kepler may have also found the elongated square gyrobicupola (pseudorhombicuboctahedron): at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, and the first clear statement of the pseudorhombicuboctahedron's existence was made in 1905, by Duncan Sommerville.

Classification
There are 13 Archimedean solids (not counting the elongated square gyrobicupola; 15 if the mirror images of two enantiomorphs, the snub cube and snub dodecahedron, are counted separately).

Here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a vertex configuration of 4.6.8 means that a square, hexagon, and octagon meet at a vertex (with the order taken to be clockwise around the vertex).

Some definitions of semiregular polyhedron include one more figure, the elongated square gyrobicupola or "pseudo-rhombicuboctahedron".

Properties
The number of vertices is 720° divided by the vertex angle defect.

The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular.

The duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices.

Chirality
The snub cube and snub dodecahedron are known as chiral, as they come in a left-handed form (Latin: levomorph or laevomorph) and right-handed form (Latin: dextromorph). When something comes in multiple forms which are each other's three-dimensional mirror image, these forms may be called enantiomorphs. (This nomenclature is also used for the forms of certain chemical compounds.)

Construction of Archimedean solids

The different Archimedean and Platonic solids can be related to each other using a handful of general constructions. Starting with a Platonic solid, truncation involves cutting away of corners. To preserve symmetry, the cut is in a plane perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners. Depending on how much is truncated (see table below), different Platonic and Archimedean (and other) solids can be created. If the truncation is exactly deep enough such that each pair of faces from adjacent vertices shares exactly one point, it is known as a rectification. An expansion, or cantellation, involves moving each face away from the center (by the same distance so as to preserve the symmetry of the Platonic solid) and taking the convex hull. Expansion with twisting also involves rotating the faces, thus splitting each rectangle corresponding to an edge into two triangles by one of the diagonals of the rectangle. The last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed as the rectification of the rectification. Likewise, the cantitruncation can be viewed as the truncation of the rectification.

Note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron. Also, partially because the tetrahedron is self-dual, only one Archimedean solid that has at most tetrahedral symmetry. (All Platonic solids have at least tetrahedral symmetry, as tetrahedral symmetry is a symmetry operation of (i.e. is included in) octahedral and isohedral symmetries, which is demonstrated by the fact that an octahedron can be viewed as a rectified tetrahedron, and an icosahedron can be used as a snub tetrahedron.)

General references

 * Chapter 2
 * (Section 3–9)
 * (Section 3–9)

title: Archimedean solid


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Talk:Archimedean solid

Untitled
"semi-regular" should be defined; right now it is not clear what the difference between Johnson and Archimedian solids is. Furthermore, I take it that the platonic solids also count as Archimedian? --Axelboldt The first graph is wrong it is nt the largest  — Preceding unsigned comment added by 212.166.157.122 (talk) 18:37, 24 November 2018 (UTC)

I don't think so. semi-regular, i think has to do with the fact that multiple kinds polygons can meet at a vertex And platonic solids are definitely not Archimedean. (since there are five platonic solids, and thirteen archimedean ones....)

 * I think not being face-uniform just means there are more than one kind of regular polygon used. And I guess that being vertex-uniform means that there is a rotation that can move any one chosen vertex to the place of any other chosen vertex, while also mapping all other verticies to the place of either itself or another vertex. I hope so, because if I guessed wrongly, then my pictures and coordinates probably aren't of Archimedean solids. Ксйп Cyp 20:24, 31 Jul 2003 (UTC)

 * Hey, that was a reply in less than 2 years... How's that for wiki-fast replies? Ксйп Cyp 23:16, 31 Jul 2003 (UTC)

 * Suppose to the above comment on rotation, mirroring also counts, since otherwise I can't see how a truncated cuboctahedron or truncated icosidodecahedron would count. Ксйп Cyp 10:13, 1 Aug 2003 (UTC)

A platonic solid is made up of only one type of regular polygon, and is vertex transitive. An Archimedian solid is any vertex transitive solid and is made up of two or more regular polygons. Obviously, these two can't be the same. And a Johnson solid is any solid that is made up of two or more regular polygons, that aren't vertex transitive. Vertex transitive means that any vertex, all the same polygons meet, in the same order and number. Hope that clears it up. Timeroot (talk) 00:33, 24 June 2008 (UTC)

 * Not quite. Vertex-transitivity means that the solid has a symmetry group under which any vertex maps to any other.  The congruency of vertex figures follows from this, but not the other way around.  The elongated square gyrobicupola (aka "false rhombicuboctahedron") has a triangle and three squares at each vertex, but the sixteen vertices of its octagonal prism are not in the same symmetry class as the eight vertices of the 'tops' of the cupolae.  —Tamfang (talk) 04:02, 9 July 2008 (UTC)

The last sentence uses the term "regular vertex" without defining it. AxelBoldt 09:50, 2 Oct 2003 (UTC)

I worked a bit on the definition of Archimedean solids, but it's still not satisfactory. Right now, it's not clear why the Elongated Square Gyrobicupola is excluded. AxelBoldt 10:13, 2 Oct 2003 (UTC)

Found http://www.math.washington.edu/~king/coursedir/m444a03/as/polyhedra-links.html

Contains a lot of sections, seems to be in the format of mentioning an Archimedean solid site, someone reviewing it, and selecting their favourite solid. There's at least 2 about this page. I hope copying the comments about this site to here would be considered fair use or something... Κσυπ Cyp    19:15, 18 Jan 2004 (UTC)

Jennifer Brosten reviews http://en2.wikipedia.org/wiki/Archimedean_solid

This website was nice because it has the general idea of the Archimedean solids in a rather concise manor though it also offers more in depth information on polyhedras and all of the different Archimedean Polyhedra. The table showing the different Archimedean Polyhedra was well done, because it illustrated the figures while also giving useful information about the vertices, faces and how they meet. The illustrations make it so that you can see the 3D aspect, whereas many sites only show the front without being able to see what is happening at the back of the object as well. The web site does not stop at giving the general information of the different solids. If you click on the names of the solids, you are taken to a new web page which is devoted strictly to that solid.

My favorite Archimedean polyhedra would have to be the Icosidodecahedron. What made me pick this shape was first it's name, because it's kinda fun to try to say. The Icosidodecahedron is made up of 20 triangular faces and 12 pentagon faces. There are a total of 60 edges and 30 vertices. At each vertex, there are 2 triangles and 2 pentagons meeting. They go triangle-pentagon-triangle-pentagon.

Mine would have to be one with both icosahedral-group and octahedral-group symmetry. Does any such polyhedron, Archimedean or not, exist? —The Doctahedron, 68.173.113.106 (talk) 21:50, 22 November 2011 (UTC)

Mary Moser reviews http://www.ezresult.com/article/Archimedean_solid  (Note: That is a not so good mirror of Wikipedia, without working pictures... Ironically, the only complaint about the article is that the pictures don't work.)

I was able to find the above website and saw that it had all the basic information presented clearly as well as some interesting history. I really appreciated that throughout the sight vocab words are linked to further definitions and explenations. There is a lot of potential in this sight unfortunately it seems the pages that are supposed to provide images of the polyhedra are not working, (at least I was unable to view them). Also I would like to see descriptions relating them to the platonic solids (essentially how we get the Archimedean polyhedron by truncating the platonic polyhedron).

I think a good example of the pictures and information my initial sight is missing can be found at http://www.ul.ie/~cahird/polyhedronmode/favorite.htm. I especially liked the animation showing the truncations for some of the polyhedra.

While each of the polyhedra we are discussing is really interesting and fun to explore, the assignment is to choose one favorite so I choose truncated cuboctahedron. It has 26 faces (12 squares, 8 hexagons and 6 octagons), 72 edges, and 48 vertices.

Educational toy
Jovo [] is a toy that is ideal for constructing Archimedean solids. Can such a link be in the Wikipedia, or is it too commercial? --80.162.63.207 17:01, 5 Feb 2005 (UTC)

Inscribed Archimedean polyhedra
NOTE: This text for a new and uncompleted section was removed from article!
 * Edit here as you like and move back when it's done!
 * Tom Ruen 21:43, 14 January 2006 (UTC)

When the Archimedean polyhedra are inscribed in a sphere, they occupy the following percentages of that sphere's volume:
 * truncated tetrahedron 40.134%
 * cuboctahedron 56.270%
 * truncated cube 57.682%
 * truncated octahedron XX%
 * rhombicuboctahedron XX%
 * truncated cuboctahedron XX%
 * snub cube XX%
 * icosidodecahedron XX%
 * truncated dodecahedron XX%
 * truncated icosahedron XX%
 * rhombicosidodecahedron XX%
 * truncated icosidodecahedron XX%
 * snub dodecahderon XX%

 * I hope someone who has the percentages will add them here. I was attempting to make the Archimedean article include some of the same things that are in the Platonic article. Unfortunately, removing the list to the discussion page instead of leaving it on the article page means that it will probably languish here, and not get finished, but we'll see. Jimaginator 13:06, 19 January 2006 (UTC)

Here is how you could calculate those percentages: Go onto the page for that solid, and scroll to the part about the cartesian co-ordinates. Take the three numbers, square each of them, and take their sum (for the truncated tetrahedron, these would yield 11, for example). Now take the square root of that. Now, in that same section, it will say "For a side length of --." The truncated tetrahedron, for example, will have "the square root of 8". Divide this side length number by the previous number you got (the square root of 11), and put that into the formula for volume, which will be else where on the page. Now, divide that result of that formula by 4/3pi, and multiply by 100. This is your percentage, and it will work for any Archimedean solid. I already did a few. Timeroot (talk) 17:57, 24 June 2008 (UTC)

Wythoff construction
All the Archimedean solids can be constructed by Wythoff construction of a spherical tiling.

Blue and yellow
Hi, I've just created the following table, and I think the bottom line would look much more logical with blue and yellow exchanged. The trigons and hexagons, which are blue at the moment, correspond to the yellow trigons in the two big files above. The squares (or trigon pairs in the snubs), yellow at the moment, have no corresponding faces in all the files above. They just correspond to the vertices in the two big files. Thus I think, these "new" faces should have the "new" color. Any thoughts? Watchduck (talk) 23:33, 7 August 2010 (UTC)

Yes. —Tamfang (talk) 03:01, 9 August 2010 (UTC)

That would make sense. Who made these pictures, anyway? —The Doctahedron, 68.173.113.106 (talk) 21:45, 22 November 2011 (UTC)

With almost 8 years delay I have done that now. Watchduck (quack) 01:42, 16 April 2018 (UTC)

Why are there only 13 Archimedean solids?
Could someone add a section on why there are only 13 Archimedean solids please, and explain it well? Thanks, please do it asap Akhi666 (talk) 18:55, 11 October 2014 (UTC)

Figurate numbers?
Just as platonic solids have formulas for generating figurate numbers Figurate_number, are there any formulas for generating numbers of Archimedean solids? — Preceding unsigned comment added by R3hall (talk • contribs) 05:47, 30 April 2016 (UTC)

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Images
I propose replacing the solids and vertex figures (and nets) by more consistent images. This is partly the same topic I already brought up in the Blue and yellow section above. I made the proposed changes in the article (edit) and in the template (edit) where the images for the individual articles come from. The old vertex figure images show the complete faces around the vertex. I don't think that is necessary when they are shown in context with the full solids — which is always the case in these articles. And the vertex figure is a single polygon after all.

I request that you do not immediately revert my changes here and in the template, and instead leave them visible for a while, so others can see them and offer their opinion. I am quite sure that you will be against this change, but it would be good to get more than two opinions on this. (Apparently that was already too much to ask for.)

I found it absurd to waste vertical space in the transparent column for these animation links, so I did that thing with the gear icons. It's just a proposal, and even someone who finds it horrible should not find it horrible enough to revert my edit because of that. Just say so below, and a better solution will be found. Watchduck (quack) 01:44, 16 April 2018 (UTC)

 * The partial polygon faces are confusing. There's nothing to argue here. That's a fact. Tom Ruen (talk) 01:48, 16 April 2018 (UTC)

 * A wider discussion on illustrating vertex figures in general has now been started at Talk:Vertex figure. I have commented there and posted some example diagrams. — Cheers, Steelpillow (Talk) 11:32, 16 April 2018 (UTC)

 * Vertex figure images similar to the one shown on the right are among the possible alternatives. See the general discussion mentioned above. Watchduck (quack) 17:45, 16 April 2018 (UTC)

 * Compare the illustration of cuboctahedron and rhombicosidodecahedron vertex figures in Figure 21.1 (p. 289) in The Symmetries of Things by Conway et al. Watchduck (quack) 23:30, 26 July 2018 (UTC)

 * This is also dealt with in the linked discussion. — Cheers, Steelpillow (Talk) 17:58, 27 July 2018 (UTC)

 * (Comment by Tomruen copied here by Watchduck from the linked discussion, because my answer is specific to this article.)
 * TSoT also shows vertex figures with number labels for each p-gon. Tom Ruen (talk) 00:40, 27 July 2018 (UTC)

 * I am not opposed to including labels. How about something like the image on the right? (Shown as 140px like in the infobox.) Watchduck (quack) 16:35, 27 July 2018 (UTC)

 * The following reply of mine also needed to be copied across from the other discussion because it is also relevant here.
 * You are cherry-picking snippets to try and back your case. The text in TSOT accompanying the illustrations on p.289 states categorically that "the vertex figure of the cuboctahedron (Figure 21.1 left) and icosidodecahedron are rectangles whose sides are numbered ...". The drawing shows a slice through a polyhedron, the "pyramid" is merely that part to one side of the slice, with the overall polyhedral form remaining intact. It does not, as you claim, merely "show a pyramid cut from the solid". Note the close correspondence between Fig 21.1 and the cuboctahedron drawing I mentioned in the other discussion.
 * I am not sure if polychora have been mentioned here, but if a pyramid does occur as a vertex figure then it is of some four-dimensional polychoron. It is, as I have said, mathematical madness to use the same pyramid in an attempt to illustrate the lower-dimensional vertex figure of a polyhedron. The central point adds nothing to the description here, save confusion.— Cheers, Steelpillow (Talk) 20:19, 27 July 2018 (UTC)

 * No need to outdent. I doubt this belongs here, because it's not specific to Archimedean solids. With your quote from TSoT ("states categorically" etc.) you imply that our disagreement is factual, and not just stylistic. I consider this a straw man done in bad faith, because in the other discussion I have already made clear that we do not disagree about what a vertex figure is ("not proposing to illustrate a pyramid" etc.). Watchduck (quack) 21:13, 27 July 2018 (UTC)
 * The problem is that when the central vertex is included, what the image shows is indeed a pyramid, regardless of whether or not it is proposing to illustrate that. Double sharp (talk) 09:32, 18 September 2018 (UTC)

This should be solved now. I created a set of vector files that show the full faces and the outline of the vertex figure. 01:58, 10 December 2020 (UTC)

Capitalization
About colons, Manual of Style/Capital letters says, "The same {not capitalizing the word immediately following a punctuation mark} usually applies after colons, although sometimes the word following a colon is capitalized, if that word effectively begins a new grammatical sentence, and especially if the colon serves to introduce more than one sentence." I think that the length of the sentence that follows the colon warrants capitalization. Also, while the section does not mention parentheses, the sentence inside the parentheses is a complete sentence, so I assume that it should be capitalized. Care to differ or discuss with me? The Nth User 02:36, 4 November 2018 (UTC)
 * You have an arguable point concerning the colon and perhaps a better solution would be to eliminate the colon altogether as the sentence containing it is borderline run-on. As for the parenthetical remark I would point to the Sentences and brackets section of MOS:PAREN which states, "A sentence that occurs within brackets in the course of another sentence does not generally have its first word capitalized nor end with a period (full stop) just because it starts a sentence". --Bill Cherowitzo (talk) 04:50, 4 November 2018 (UTC)
 * I didn't see the information at MOS:PAREN in Manual of Style/Capital letters, so maybe it should be added, but now that I know that the MoS has a guideline about it, I am going to go by that. As for the colon, I agree that the sentence should be broken up, but I am not sure that the colon is the best place to do it. Even if it was broken at the colon, the sentence would still be somewhat long."Some authors give a weaker definition of an Archimedean solid, in which 'identical vertices' means merely that the faces surrounding each vertex are of the same types (each vertex looks the same from close up), so only a local isometry is required, but then they omit a 14th polyhedron that meets this weaker definition, elongated square gyrobicupola (or pseudo-rhombicuboctahedron), but not the stronger definition and only list the 13 Archimedean solids."The obvious place to split this sentence is at the switch from discussing which isometry definition authors use to discussing whether or not authors include the pseudo-rhombicuboctahedron. This results in two sentences of about the same length."Some authors give a weaker definition of an Archimedean solid, in which 'identical vertices' means merely that the faces surrounding each vertex are of the same types (each vertex looks the same from close up), so only a local isometry is required. However, these authors omit a 14th polyhedron that meets this weaker definition, elongated square gyrobicupola (or pseudo-rhombicuboctahedron), but not the stronger definition and only list the 13 Archimedean solids."Because the part before the colon isn't that long, the colon probably doesn't have to be replaced with a period. Keeping the colon, moving the naming of the polyhedron fowards a few words, and adding an i.e. to the beginning of the statement in the parentheses results in this."pointed out a widespread error in the literature on Archimedean solids: some authors give a weaker definition of an Archimedean solid, in which 'identical vertices' means merely that the faces surrounding each vertex are of the same types (i.e. each vertex looks the same from close up), so only a local isometry is required. However, these authors omit a 14th polyhedron, the elongated square gyrobicupola (or pseudo-rhombicuboctahedron), that meets this weaker definition but not the stronger definition and only list the 13 Archimedean solids."Incidentally, because the colon would now be introducing two sentences (because neither the fact that most authors use the weaker definition nor the fact that most authors do not include the pseudo-rhombicuboctahedron amounts to an inconsistency or contradiction individually), there would be an even stronger case for capitalizing the word some. Care to differ or discuss with me? The Nth User 04:16, 6 November 2018 (UTC)
 * These are definite improvements, but I still think that this can be done without a colon. For example,

"pointed out that some authors give a weaker definition of an Archimedean solid, in which 'identical vertices' means merely that the faces surrounding each vertex are of the same types (i.e. each vertex looks the same from close up), so only a local isometry is required. Grünbaum decries a common error in the literature, namely, these authors omit a 14th polyhedron, the elongated square gyrobicupola (or pseudo-rhombicuboctahedron), that meets this weaker definition but not the stronger definition and thus only list the 13 Archimedean solids." --Bill Cherowitzo (talk) 16:45, 6 November 2018 (UTC)
 * As a whole, I like your version, but there are some things that I would change, like making the point of the first sentence more obvious sooner and removing the word decries, which I think is too strong. How about this?"pointed out an error where some authors give a weaker definition of an Archimedean solid, in which 'identical vertices' means merely that the faces surrounding each vertex are of the same types (i.e. each vertex looks the same from close up), so only a local isometry is required. The contradiction arises when these authors omit a 14th polyhedron, the elongated square gyrobicupola (or pseudo-rhombicuboctahedron), that only meets this weaker definition. In order to be consistent, the authors who omit the pseudo-rhombicuboctahedron (and most do) must use the stronger, global isometry-based definition instead of the weaker, local isometry based definition."I think that summarizing the inconsistency in a way that offers a resolution to the problem. Care to differ or discuss with me? The Nth User 04:07, 7 November 2018 (UTC)
 * We are getting closer, but I do have some problems with this version. First of all, it is not an error to use a weaker definition, the error only arises if you use the weaker definition and don't include the 14th example. Also, this is not a contradiction, it is a bona fide error. I used "decries" only because I didn't want to repeat the use of "points out" in two adjacent sentences, any other wording would suffice. --Bill Cherowitzo (talk) 19:40, 7 November 2018 (UTC)
 * How about something like
 * observed that a 14th polyhedron, the elongated square gyrobicupola (or pseudo-rhombicuboctahedron), meets a weaker definition of an Archimedean solid, in which "identical vertices" means merely that the faces surrounding each vertex are of the same types (i.e. each vertex looks the same from close up), so only a local isometry is required. Grünbaum pointed out a frequent error in which authors define Archimedean solids using this local definition but omit the 14th polyhedron. If only 13 polyhedra are to be listed, the definition must use global symmetries of the polyhedron rather than local neighborhoods.
 * ? —David Eppstein (talk) 19:54, 7 November 2018 (UTC)
 * Thanks David, that looks good to me. --Bill Cherowitzo (talk) 20:11, 7 November 2018 (UTC)
 * My only suggestion is that the last sentence be rephrased to If the weaker, "local" definition is to be used, the 14th polyhedron, the pseduo-rhombicuboctahedron, must be included. because that is what Grünbaum has suggested. Care to differ or discuss with me? The Nth User 01:23, 9 November 2018 (UTC)
 * I don't really feel strongly about this, but I chose the contrapositive formulation because our article generally takes the position that there are only 13, despite what Grünbaum said. His suggestion to use 14 and the weaker definition have not been widely accepted. So your formulation looks like "if counterfactual thing X, then counterfactual thing Y" (a vacuous truth) while the way I wrote it looks like "if (thing we already say is true in the article), then (other thing we already say is true in the article)". They're logically equivalent but I think it's less confusing that way. —David Eppstein (talk) 01:40, 9 November 2018 (UTC)
 * You have a valid point. What about this?"observed that a 14th polyhedron, the elongated square gyrobicupola (or pseudo-rhombicuboctahedron), meets a weaker definition of an Archimedean solid, in which 'identical vertices' means merely that the faces surrounding each vertex are of the same types (i.e. each vertex looks the same from close up), so only a local isometry is required. Grünbaum pointed out a frequent error in which authors define Archimedean solids using this local definition but omit the 14th polyhedron. Grünbaum suggests that since the weaker, 'local' definition is used more often, the pseduo-rhombicuboctahedron should be counted as an Archimedean solid. However, most authors only list 13 polyhedra, which means that they must use the definition that utilizes global symmetries of the polyhedron rather than local neighborhoods." Care to differ or discuss with me? The Nth User 04:04, 10 November 2018 (UTC)
 * Your version is internally inconsistent. It says that the local version is used more often, and then it says that most authors must use the global definition. Which is it, and what is the evidence for the claim about the frequency of use of different definitions? —David Eppstein (talk) 04:22, 10 November 2018 (UTC)
 * I think that that is the error that Grünbaum points out: Most authors use the weaker definition but omit the pseudorhombicuboctahedron. Please tell me if that is not what you mean. Care to differ or discuss with me? The Nth User 04:30, 10 November 2018 (UTC)
 * If your last sentence says that most authors should and do use the global definition, it contradicts the earlier sentence that says that most authors use the local definition. If it says that most authors should use the global definition (but don't), it is an inappropriate insertion of opinion, because it would be equally valid for those authors to switch to the 14-polyhedron count and stick with the local version. And in any case, "most" means "more than half" and would need to be backed by a reliable source that actually surveys authors and counts how many of them follow each variant, which we don't have. —David Eppstein (talk) 04:55, 10 November 2018 (UTC)
 * My last sentence means that must authors do not include the pseudorhombicuboctahedron, so they must use the stronger definition for consistency, while the earlier sentence says that Grünbaum thinks that all authors should include the pseudorhombicuboctahedron. Neither of these is my opinion. Also, the article already says that most authors omit the pseudorhombicuboctahedron. Care to differ or discuss with me? The Nth User 05:02, 10 November 2018 (UTC)
 * You're still falling into the trap of dictating to authors which solution they should take rather than merely pointing out that some of them got it wrong. —David Eppstein (talk) 05:55, 10 November 2018 (UTC)
 * In that case, just remove the last sentence and put the rest in. Care to differ or discuss with me? The Nth User 23:49, 10 November 2018 (UTC)

User:The Nth User/Extended Archimedean solid table
What do you think? I think that it helps to show underlying pattern and definitely belongs somewhere, but not necessarily here. Care to differ or discuss with me? The Nth User 03:44, 18 December 2018 (UTC)

Snub Cube Errata
In the "Classification" data table for the 13 solids, the Volume shown for Snub Cube, 7.889295, is incorrect. The correct value, 7.8894774, is available at various websites. My own calculation with standard precision Excel comes to 7.88947739998, so 7.8894774 is a deceptively nice approximation. The Sphericity value, =(pi*(6V)^2)^(1/3)/A, appears to have been calculated with correct surface area, A, and the incorrect volume listed. Consequently, the Sphericity listed, .9651814, is also incorrect. I have snub cube area as 19.85640646 and sphericity as .96519625.

I am confident of this errata but would prefer one of the primary authors confirm and make the change. Change Snub Cube volume, was 7.889295, to 7.8894774. Change Snub Cube sphericity, was .9651814, to .9651963. JFSather (talk) 19:15, 19 October 2021 (UTC)Jim Sather

Johnson

 * The Archimedean solids are a subset of the Johnson solids,

Oh? Usually the latter are defined as strictly convex polyhedra with regular faces that are not uniform. —Tamfang (talk) 04:45, 13 September 2023 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Archimedean solid


text:
Antiprism

In geometry, an $n$-gonal antiprism or $n$-antiprism is a polyhedron composed of two parallel direct copies (not mirror images) of an $n$-sided polygon, connected by an alternating band of $n = 6$ triangles. They are represented by the Conway notation $2n$.

Antiprisms are a subclass of prismatoids, and are a (degenerate) type of snub polyhedron.

Antiprisms are similar to prisms, except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are $4n$ triangles, rather than $n$ quadrilaterals.

The dual polyhedron of an $n$-gonal antiprism is an $n$-gonal trapezohedron.

History
In his 1619 book Harmonices Mundi, Johannes Kepler observed the existence of the infinite family of antiprisms. This has conventionally been thought of as the first discovery of these shapes, but they may have been known earlier: an unsigned printing block for the net of a hexagonal antiprism has been attributed to Hieronymus Andreae, who died in 1556.

The German form of the word "antiprism" was used for these shapes in the 19th century; Karl Heinze credits its introduction to Theodor Wittstein. Although the English "anti-prism" had been used earlier for an optical prism used to cancel the effects of a primary optimal element, the first use of "antiprism" in English in its geometric sense appears to be in the early 20th century in the works of H. S. M. Coxeter.

Right antiprism
For an antiprism with regular $n$-gon bases, one usually considers the case where these two copies are twisted by an angle of $2n$ degrees.

The axis of a regular polygon is the line perpendicular to the polygon plane and lying in the polygon centre.

For an antiprism with congruent regular $n$-gon bases, twisted by an angle of $3.3.3.n$ degrees, more regularity is obtained if the bases have the same axis: are coaxial; i.e. (for non-coplanar bases): if the line connecting the base centers is perpendicular to the base planes. Then the antiprism is called a right antiprism, and its ${ }⊗{n}$ side faces are isosceles triangles.

Uniform antiprism
A uniform $n$-antiprism has two congruent regular $n$-gons as base faces, and $s{2,2n} sr{2,n}$ equilateral triangles as side faces.

Uniform antiprisms form an infinite class of vertex-transitive polyhedra, as do uniform prisms. For $An$, we have the digonal antiprism (degenerate antiprism), which is visually identical to the regular tetrahedron; for $Dnd, [2+,2n], (2*n)$, the regular octahedron as a triangular antiprism (non-degenerate antiprism).

The Schlegel diagrams of these semiregular antiprisms are as follows:

Cartesian coordinates
Cartesian coordinates for the vertices of a right $n$-antiprism (i.e. with regular $n$-gon bases and $4n$ isosceles triangle side faces) are:
 * $$\left( \cos\frac{k\pi}{n}, \sin\frac{k\pi}{n}, (-1)^k h \right)$$

where $Dn, [2,n]+, (22n)$;

if the $n$-antiprism is uniform (i.e. if the triangles are equilateral), then: $$2h^2 = \cos\frac{\pi}{n} - \cos\frac{2\pi}{n}.$$

Volume and surface area
Let $n$ be the edge-length of a uniform $n$-gonal antiprism; then the volume is: $$V = \frac{n\sqrt{4\cos^2\frac{\pi}{2n}-1}\sin \frac{3\pi}{2n} }{12\sin^2\frac{\pi}{n}}~a^3,$$

and the surface area is: $$A = \frac{n}{2} \left( \cot\frac{\pi}{n} + \sqrt{3} \right) a^2.$$

Furthermore, the volume of a regular right $n$-gonal antiprism with side length of its bases $a$ and height $n$ is given by: $$V = \frac{nhl^2}{12} \left( \csc\frac{\pi}{n} + 2\cot\frac{\pi}{n}\right).$$

Note that the volume of a right $n$-gonal prism with the same $l$ and $h$ is: $$V_{\mathrm{prism}}=\frac{nhl^2}{4} \cot\frac{\pi}{n}$$ which is smaller than that of an antiprism.

Symmetry
The symmetry group of a right $n$-antiprism (i.e. with regular bases and isosceles side faces) is $2n$ of order $n = 9$, except in the cases of:
 * $2n$: the regular tetrahedron, which has the larger symmetry group $An$ of order $2n$, which has three versions of $180⁄n$ as subgroups;

 * $180⁄n$: the regular octahedron, which has the larger symmetry group $2n$ of order $2n$, which has four versions of $n = 2$ as subgroups.

The symmetry group contains inversion if and only if $l$ is odd.

The rotation group is $n = 3$ of order $2n$, except in the cases of:
 * $0 ≤ k ≤ 2n – 1$: the regular tetrahedron, which has the larger rotation group $Dnd = Dnv$ of order $4n$, which has three versions of $n = 2$ as subgroups;

 * $T_{d}$: the regular octahedron, which has the larger rotation group $24 = 3 × (4 × 2)$ of order $D_{2d}$, which has four versions of $n = 3$ as subgroups.

Note: The right $h$-antiprisms have congruent regular $n$-gon bases and congruent isosceles triangle side faces, thus have the same (dihedral) symmetry group as the uniform $n$-antiprism, for $O_{h}$.

In higher dimensions
Four-dimensional antiprisms can be defined as having two dual polyhedra as parallel opposite faces, so that each three-dimensional face between them comes from two dual parts of the polyhedra: a vertex and a dual polygon, or two dual edges. Every three-dimensional convex polyhedron is combinatorially equivalent to one of the two opposite faces of a four-dimensional antiprism, constructed from its canonical polyhedron and its polar dual. However, there exist four-dimensional polyhedra that cannot be combined with their duals to form five-dimensional antiprisms.

Self-crossing polyhedra

Uniform star antiprisms are named by their star polygon bases, $48 = 4 × (4 × 3)$ and exist in prograde and in retrograde (crossed) solutions. Crossed forms have intersecting vertex figures, and are denoted by "inverted" fractions: $D_{3d}$ instead of $D_{n}$; example: 5/3 instead of 5/2.

A right star antiprism has two congruent coaxial regular convex or star polygon base faces, and $2n$ isosceles triangle side faces.

Any star antiprism with regular convex or star polygon bases can be made a right star antiprism (by translating and/or twisting one of its bases, if necessary).

In the retrograde forms, but not in the prograde forms, the triangles joining the convex or star bases intersect the axis of rotational symmetry. Thus:

 * Retrograde star antiprisms with regular convex polygon bases cannot have all equal edge lengths, and so cannot be uniform. "Exception": a retrograde star antiprism with equilateral triangle bases (vertex configuration: 3.3/2.3.3) can be uniform; but then, it has the appearance of an equilateral triangle: it is a degenerate star polyhedron.

 * Similarly, some retrograde star antiprisms with regular star polygon bases cannot have all equal edge lengths, and so cannot be uniform. Example: a retrograde star antiprism with regular star 7/5-gon bases (vertex configuration: 3.3.3.7/5) cannot be uniform.

Also, star antiprism compounds with regular star $n = 2$-gon bases can be constructed if $n$ and $n$ have common factors. Example: a star 10/4-antiprism is the compound of two star 5/2-antiprisms.

title: Antiprism


text:
Natural history of Africa

The natural history of Africa encompasses some of the well known megafauna of that continent.

Natural history is the study and description of organisms and natural objects, especially their origins, evolution, and interrelationships.

Flora
The vegetation of Africa follows very closely the distribution of heat and moisture. The northern and southern temperate zones have a flora distinct from that of the continent generally, which is tropical. In the countries bordering the Mediterranean, there are groves of orange and olive trees, evergreen oaks, cork trees and pines, intermixed with cypresses, myrtles, arbutus and fragrant tree-heaths.

South of the Atlas Mountains, the conditions alter. The zones of minimum rainfall have a very scanty flora, consisting of plants adapted to resist the great dryness. Characteristic of the Sahara is the date palm, which flourishes where other vegetation can scarcely maintain existence, while in the semidesert regions the acacia, from which gum arabic is obtained, is abundant.

The more humid regions have a richer vegetation; dense forest where the rainfall is greatest and variations of temperature least, conditions found chiefly on the tropical coasts, and in the west African equatorial basin with its extension towards the upper Nile; and savanna interspersed with trees on the greater part of the plateaus, passing as the desert regions are approached into a scrub vegetation consisting of thorny acacias, etc. Forests also occur on the humid slopes of mountain ranges up to a certain elevation. In the coast regions, the typical tree is the mangrove, which flourishes wherever the soil is of a swamp character.

The dense forests of West Africa contain, in addition to a great variety of hardwoods, two palms, Elaeis guineensis (oil palm) and Raphia vinifera (bamboo palm), not found, generally speaking, in the savanna regions. Bombax or silk cotton trees attain gigantic proportions in the forests, which are the home of the India rubber-producing plants and of many valuable kinds of timber trees, such as odum (Chlorophora excelsa), ebony, mahogany (Khaya senegalensis), Oldfieldia (Oldfieldia africana) and camwood (Baphia nitida). The climbing plants in the tropical forests are exceedingly luxuriant and the undergrowth or "bush" is extremely dense.

In the savannas the most characteristic trees are the monkey-bread tree or baobab (Adansonia digitata), doum palm (Hyphaene) and euphorbias. The coffee plant grows wild in such widely separated places as Liberia and southern Ethiopia. The higher mountains have a special flora showing close agreement over wide intervals of space, as well as affinities with the mountain flora of the eastern Mediterranean, the Himalaya and Indo-China.

In the swamp regions of north-east Africa, papyrus and associated plants, including the soft-wooded ambach, flourished in immense quantities, and little else is found in the way of vegetation. South Africa is largely destitute of forest, save in the lower valleys and coast regions. Tropical flora disappears, and in the semi-desert plains the fleshy, leafless, contorted species of kapsias, mesembryanthemums, aloes and other succulent plants make their appearance. There are, too, valuable timber trees, such as the yellowwood (Podocarpus elongatus), stinkwood (Ocotea), sneezewood or Cape ebony (Pteroxylon utile) and ironwood. Extensive miniature woods of heaths are found in almost endless variety and covered throughout the greater part of the year with innumerable blossoms in which red is very prevalent. Of the grasses of Africa, alfa is very abundant in the plateaus of the Atlas range.

Fauna
The fauna again shows the effect of the characteristics of the vegetation. The open savannas are the home of large ungulates, especially antelopes, the giraffe (peculiar to Africa), zebra, buffalo, wild donkey and four species of rhinoceros; and of carnivores, such as the lion, leopard, hyena, etc. The okapi (a genus restricted to Africa) is found only in the dense forests of the Congo basin. Bears are confined to the Atlas region, wolves and foxes to North Africa. The elephant (though its range has become restricted through the attacks of hunters) is found both in the savannas and forest regions, the latter being otherwise poor in large game, though the special habitat of the chimpanzee and gorilla. Baboons and mandrills, with few exceptions, are peculiar to Africa. The single-humped camel, as a domestic animal, is especially characteristic of the northern deserts and steppes.

The rivers in the tropical zone abound with hippopotami and crocodiles, the former entirely confined to Africa. The vast herds of game, formerly so characteristic of many parts of Africa, have much diminished with the increase of intercourse with the interior. Game reserves have, however, been established in South Africa, Central Africa, East Africa, Somaliland, etc., while measures for the protection of wild animals were laid down in an international convention signed in May 1900.

The ornithology of northern Africa presents a close resemblance to that of southern Europe, scarcely a species being found which does not also occur in the other countries bordering the Mediterranean. Among the birds most characteristic of Africa are the ostrich and the secretarybird. The ostrich is widely dispersed, but is found chiefly in the desert and steppe regions. The secretarybird is common in the south. The weaver birds and their allies, including the long-tailed whydahs, are abundant, as are, among game-birds, the francolin and guineafowl. Many of the smaller birds, such as the sunbirds, bee-eaters, the parrots and kingfishers, as well as the larger plantain-eaters, are noted for the brilliance of their feathers.

Of reptiles, the lizard and chameleon are common, and there are a number of venomous snakes, though these are not so numerous as in other tropical countries.

The scorpion is abundant. Of insects, Africa has many thousand different kinds; of these the locust is the proverbial scourge of the continent, and the ravages of the termites are almost incredible. The spread of malaria by means of mosquitoes is common. The tsetse fly, whose bite is fatal to all domestic animals, is common in many districts of South and East Africa. It is found nowhere outside Africa.

title: Natural history of Africa


text:
Geography of Africa

Africa is a continent comprising 63 political territories, representing the largest of the great southward projections from the main mass of Earth's surface. Within its regular outline, it comprises an area of 30368609 km2, excluding adjacent islands. Its highest mountain is Kilimanjaro; its largest lake is Lake Victoria.

Separated from Europe by the Mediterranean Sea and from much of Asia by the Red Sea, Africa is joined to Asia at its northeast extremity by the Isthmus of Suez (which is transected by the Suez Canal), 130 km wide. For geopolitical purposes, the Sinai Peninsula of Egypt – east of the Suez Canal – is often considered part of Africa. From the most northerly point, Ras ben Sakka in Tunisia, at 37°21′ N, to the most southerly point, Cape Agulhas in South Africa, 34°51′15″ S, is a distance approximately of 8000 km; from Cap-Vert, 17°31′13″W, the westernmost point, to Ras Hafun in the Somali Puntland region, in the Horn of Africa, 51°27′52″ E, the most easterly projection, is a distance (also approximately) of 7400 km.

The main structural lines of the continent show both the east-to-west direction characteristic, at least in the eastern hemisphere, of the more northern parts of the world, and the north-to-south direction seen in the southern peninsulas. Africa is thus mainly composed of two segments at right angles, the northern running from east to west, and the southern from north to south.

Main features

The average elevation of the continent approximates closely to 600 m above sea level, roughly near to the mean elevation of both North and South America, but considerably less than that of Asia, 950 m. In contrast with other continents, it is marked by the comparatively small area of either very high or very low ground, lands under 180 m occupying an unusually small part of the surface; while not only are the highest elevations inferior to those of Asia or South America, but the area of land over 3000 m is also quite insignificant, being represented almost entirely by individual peaks and mountain ranges. Moderately elevated tablelands are thus the characteristic feature of the continent, though the surface of these is broken by higher peaks and ridges. (So prevalent are these isolated peaks and ridges that a specialised term—Inselberg-Landschaft, island mountain landscape—has been adopted in Germany to describe this kind of country, thought to be in great part the result of wind action.)

As a general rule, the higher tablelands lie to the east and south, while a progressive diminution in altitude towards the west and north is observable. Apart from the lowlands and the Atlas Mountains, the continent may be divided into two regions of higher and lower plateaus, the dividing line (somewhat concave to the northwest) running from the middle of the Red Sea to about 6 degrees south on the west coast.

Africa can be divided into a number of geographic zones:

 * The coastal plains—often fringed seawards by mangrove swamps—never stretching far from the coast, apart from the lower courses of streams. Recent alluvial flats are found chiefly in the delta of the more important rivers. Elsewhere, the coastal lowlands merely form the lowest steps of the system of terraces that constitutes the ascent to the inner plateaus.
 * The Atlas range—orthographically distinct from the rest of the continent, being unconnected with and separated from the south by a depressed and desert area (the Sahara).

Plateau region
There are many plateaus in Africa.

The high southern and eastern plateaus, rarely falling below 600 m, have a mean elevation of about 1000 m. The South African plateau, as far as about 12° S, is bounded east, west and south by bands of high ground which fall steeply to the coasts. On this account South Africa has a general resemblance to an inverted saucer. Due south, the plateau rim is formed by three parallel steps with level ground between them. The largest of these level areas, the Great Karoo, is a dry, barren region, and a large tract of the plateau proper is of a still more arid character and is known as the Kalahari Desert.

The South African plateau is connected towards East African plateau, with probably a slightly greater average elevation, and marked by some distinct features. It is formed by a widening out of the eastern axis of high ground, which becomes subdivided into a number of zones running north and south and consisting in turn of ranges, tablelands and depressions. The most striking feature is the existence of two great lines of depression, due largely to the subsidence of whole segments of the Earth's crust, the lowest parts of which are occupied by vast lakes. Towards the south the two lines converge and give place to one great valley (occupied by Lake Nyasa), the southern part of which is less distinctly due to rifting and subsidence than the rest of the system.

Farther north the western hollow, known as the Albertine Rift, is occupied for more than half its length by water, forming the Great Lakes of Tanganyika, Kivu, Lake Edward and Lake Albert, the first-named over 400 mi long and the longest freshwater lake in the world. Associated with these great valleys are a number of volcanic peaks, the greatest of which occur on a meridional line east of the eastern trough. The eastern branch of the East African Rift, contains much smaller lakes, many of them brackish and without outlet, the only one comparable to those of the western trough being Lake Turkana or Basso Norok.

A short distance east of this rift valley is Mount Kilimanjaro – with its two peaks Kibo and Mawenzi, the latter being 5889 m, and the culminating point of the whole continent – and Mount Kenya, which is 5184 m. Hardly less important is the Ruwenzori Range, over 5060 m, which lies east of the western trough. Other volcanic peaks rise from the floor of the valleys, some of the Kirunga (Mfumbiro) group, north of Lake Kivu, being still partially active. This could cause most of the cities and states to be flooded with lava and ash.

The third division of the higher region of Africa is formed by the Ethiopian Highlands, a rugged mass of mountains forming the largest continuous area of its altitude in the whole continent, little of its surface falling below 1500 m, while the summits reach heights of 4400 m to 4550 m. This block of country lies just west of the line of the great East African Trough, the northern continuation of which passes along its eastern escarpment as it runs up to join the Red Sea. There is, however, in the centre a circular basin occupied by Lake Tsana.

Both in the east and west of the continent the bordering highlands are continued as strips of plateau parallel to the coast, the Ethiopian mountains being continued northwards along the Red Sea coast by a series of ridges reaching in places a height of 2000 m. In the west the zone of high land is broader but somewhat lower. The most mountainous districts lie inland from the head of the Gulf of Guinea (Adamawa, etc.), where heights of 1800 to 2400 m are reached. Exactly at the head of the gulf the great peak of the Cameroon, on a line of volcanic action continued by the islands to the south-west, has a height of 4075 m, while Clarence Peak, in Fernando Po, the first of the line of islands, rises to over 2700 m. Towards the extreme west the Futa Jallon highlands form an important diverging point of rivers, but beyond this, as far as the Atlas chain, the elevated rim of the continent is almost wanting.

Plains
Much of Africa is made up of plains of the pediplain and etchplain type often occurring as steps. The etchplains are commonly associated with laterite soil and inselbergs. Inselberg-dotted plains are common in Africa including Tanzania, the Anti-Atlas of Morocco, Namibia, and the interior of Angola. One of the most wideaspread plain is the African Surface, a composite etchplain occurring across much of the continent. The area between the east and west coast highlands, which north of 17° N is mainly desert, is divided into separate basins by other bands of high ground, one of which runs nearly centrally through North Africa in a line corresponding roughly with the curved axis of the continent as a whole. The best marked of the basins so formed (the Congo Basin) occupies a circular area bisected by the equator, once probably the site of an inland sea.

Running along the south of desert is the plains region known as the Sahel.

The arid region, the Sahara — the largest hot desert in the world, covering 9000000 km2 — extends from the Atlantic to the Red Sea. Though generally of slight elevation, it contains mountain ranges with peaks rising to 2400 m Bordered N.W. by the Atlas range, to the northeast a rocky plateau separates it from the Mediterranean; this plateau gives place at the extreme east to the delta of the Nile. That river (see below) pierces the desert without modifying its character. The Atlas range, the north-westerly part of the continent, between its seaward and landward heights encloses elevated steppes in places 160 km broad. From the inner slopes of the plateau numerous wadis take a direction towards the Sahara. The greater part of that now desert region is, indeed, furrowed by old water-channels.

Mountains
The mountains are an exception to Africa's general landscape. Geographers came up with the idea of "high Africa" and "low Africa" to help distinguish the difference in Geography; "high Africa" extending from Ethiopia down south to South Africa and the Cape of Good Hope while "low Africa" representing the plains of the rest of the continent. The following table gives the details of the chief mountains and ranges of the continent:

Rivers
From the outer margin of the African plateaus, a large number of streams run to the sea with comparatively short courses, while the larger rivers flow for long distances on the interior highlands, before breaking through the outer ranges. The main drainage of the continent is to the north and west, or towards the basin of the Atlantic Ocean.

To the main African rivers belong: Nile (the longest river of Africa), Congo (river with the highest water discharge on the continent) and the Niger, which flows half of its length through the arid areas. The largest lakes are the following: Lake Victoria (Lake Ukerewe), Lake Chad, in the centre of the continent, Lake Tanganyika, lying between the Democratic Republic of Congo, Burundi, Tanzania and Zambia. There is also the considerably large Lake Malawi stretching along the eastern border of Malawi. There are also numerous water dams throughout the continent: Kariba on the river of Zambezi, Asuan in Egypt on the river of Nile, and Akosombo, the continent's biggest dam on the Volta River in Ghana (Fobil 2003). The high lake plateau of the African Great Lakes region contains the headwaters of both the Nile and the Congo.

The break-up of Gondwana in Late Cretaceous and Cenozoic times led to a major reorganization of the river courses of various large African rivers including the Congo, Niger, Nile, Orange, Limpopo and Zambezi rivers.

Flowing to the Mediterranean Sea
The upper Nile receives its chief supplies from the mountainous region adjoining the Central African trough in the neighborhood of the equator. From there, streams pour eastward into Lake Victoria, the largest lake in Africa (covering over 26,000 square m.), and to the west and north into Lake Edward and Lake Albert. To the latter of these, the effluents of the other two lakes add their waters. Issuing from there, the Nile flows northward, and between the latitudes of 7 and 10 degrees north it traverses a vast marshy level, where its course is liable to being blocked by floating vegetation. After receiving the Bahr-el-Ghazal from the west and the Sobat, Blue Nile and Atbara from the Ethiopian Highlands (the chief gathering ground of the flood-water), it separates the great desert with its fertile watershed, and enters the Mediterranean at a vast delta.

Flowing to the Atlantic Ocean
The most remote head-stream of the Congo is the Chambezi, which flows southwest into the marshy Lake Bangweulu. From this lake issues the Congo, known in its upper course by various names. Flowing first south, it afterwards turns north through Lake Mweru and descends to the forest-clad basin of west equatorial Africa. Traversing this in a majestic northward curve, and receiving vast supplies of water from many great tributaries, it finally turns southwest and cuts a way to the Atlantic Ocean through the western highlands. The area of the Congo basin is greater than that of any other river except the Amazon, while the African inland drainage area is greater than that of any continent but Asia, where the corresponding area is 1000000 km2.

West of Lake Chad is the basin of the Niger, the third major river of Africa. With its principal source in the far west, it reverses the direction of flow exhibited by the Nile and Congo, and ultimately flows into the Atlantic — a fact that eluded European geographers for many centuries. An important branch, however — the Benue — flows from the southeast.

These four river basins occupy the greater part of the lower plateaus of North and West Africa — the remainder consists of arid regions watered only by intermittent streams that do not reach the sea.

Of the remaining rivers of the Atlantic basin, the Orange, in the extreme south, brings the drainage from the Drakensberg on the opposite side of the continent, while the Kunene, Kwanza, Ogowe and Sanaga drain the west coastal highlands of the southern limb; the Volta, Komoe, Bandama, Gambia and Senegal the highlands of the western limb. North of the Senegal, for over 1500 km of coast, the arid region reaches to the Atlantic. Farther north are the streams, with comparatively short courses, reaching the Atlantic and Mediterranean from the Atlas mountains.

Flowing to the Indian Ocean
Of the rivers flowing to the Indian Ocean, the only one draining any large part of the interior plateaus is the Zambezi, whose western branches rise in the western coastal highlands. The main stream has its rise in 11°21′3″ S 24°22′ E, at an elevation of 1500 m. It flows to the west and south for a considerable distance before turning eastward. All the largest tributaries, including the Shire, the outflow of Lake Nyasa, flow down the southern slopes of the band of high ground stretching across the continent from 10° to 12° S. In the southwest, the Zambezi system interlaces with that of the Taukhe (or Tioghe), from which it at times receives surplus water. The rest of the water of the Taukhe, known in its middle course as the Okavango, is lost in a system of swamps and saltpans that was formerly centred in Lake Ngami, now dried up.

Farther south, the Limpopo drains a portion of the interior plateau, but breaks through the bounding highlands on the side of the continent nearest its source. The Rovuma, Rufiji and Tana principally drain the outer slopes of the African Great Lakes highlands.

In the Horn region to the north, the Jubba and the Shebelle rivers begin in the Ethiopian Highlands. These rivers mainly flow southwards, with the Jubba emptying in the Indian Ocean. The Shebelle River reaches a point to the southwest. After that, it consists of swamps and dry reaches before finally disappearing in the desert terrain near the Jubba River. Another large stream, the Hawash, rising in the Ethiopian mountains, is lost in a saline depression near the Gulf of Aden.

Inland basins
Between the basins of the Atlantic and Indian Oceans, there is an area of inland drainage along the centre of the Ethiopian plateau, directed chiefly into the lakes in the Great Rift Valley. The largest river is the Omo, which, fed by the rains of the Ethiopian highlands, carries down a large body of water into Lake Turkana. The rivers of Africa are generally obstructed either by bars at their mouths, or by cataracts at no great distance upstream. But when these obstacles have been overcome, the rivers and lakes afford a vast network of navigable waters.

North of the Congo basin, and separated from it by a broad undulation of the surface, is the basin of Lake Chad — a flat-shored, shallow lake filled principally by the Chari coming from the southeast.

Lakes
The principal lakes of Africa are situated in the African Great Lakes plateau. The lakes found within the Great Rift Valley have steep sides and are very deep. This is the case with the two largest of the type, Tanganyika and Nyasa, the latter with depths of 800 m.

Others, however, are shallow, and hardly reach the steep sides of the valleys in the dry season. Such are Lake Rukwa, in a subsidiary depression north of Nyasa, and Eiassi and Manyara in the system of the Great Rift Valley. Lakes of the broad type are of moderate depth, the deepest sounding in Lake Victoria being under 90 m.

Besides the African Great Lakes, the principal lakes on the continent are: Lake Chad, in the northern inland watershed; Bangweulu and Mweru, traversed by the head-stream of the Congo; and Lake Mai-Ndombe and Ntomba (Mantumba), within the great bend of that river. All, except possibly Mweru, are more or less shallow, and Lake Chad appears to be drying up.

Divergent opinions have been held as to the mode of origin of the African Great Lakes, especially Tanganyika, which some geologists have considered to represent an old arm of the sea, dating from a time when the whole central Congo basin was under water; others holding that the lake water has accumulated in a depression caused by subsidence. The former view is based on the existence in the lake of organisms of a decidedly marine type. They include jellyfish, molluscs, prawns, crabs, etc.

Islands
With the exception of Madagascar, the African islands are small. Madagascar, with an area of 587041 km2, is, after Greenland, New Guinea and Borneo, the fourth largest island on the Earth. It lies in the Indian Ocean, off the southeast coast of the continent, from which it is separated by the deep Mozambique Channel, 400 km wide at its narrowest point. Madagascar in its general structure, as in flora and fauna, forms a connecting link between Africa and southern Asia. East of Madagascar are the small islands of Mauritius and Réunion. There are also islands in the Gulf of Guinea on which lies the Republic of São Tomé and Príncipe (islands of São Tomé and Príncipe). Part of the Republic of Equatorial Guinea is lying on the island of Bioko (with the capital Malabo and the town of Lubu) and the island of Annobón. Socotra lies E.N.E. of Cape Guardafui. Off the north-west coast are the Canary and Cape Verde archipelagoes. which, like some small islands in the Gulf of Guinea, are of volcanic origin. The South Atlantic Islands of Saint Helena and Ascension are classed as Africa but are situated on the Mid-Atlantic Ridge halfway to South America.

Climatic conditions
Lying almost entirely within the tropics, and equally to north and south of the equator, Africa does not show excessive variations of temperature.

Great heat is experienced in the lower plains and desert regions of North Africa, removed by the great width of the continent from the influence of the ocean, and here, too, the contrast between day and night, and between summer and winter, is greatest. (The rarity of the air and the great radiation during the night cause the temperature in the Sahara to fall occasionally to freezing point.) Farther south, the heat is to some extent modified by the moisture brought from the ocean, and by the greater elevation of a large part of the surface, especially in East Africa, where the range of temperature is wider than in the Congo basin or on the Guinea coast. In the extreme north and south the climate is a warm temperate one, the northern countries being on the whole hotter and drier than those in the southern zone; the south of the continent being narrower than the north, the influence of the surrounding ocean is more felt.

The most important climatic differences are due to variations in the amount of rainfall. The wide heated plains of the Sahara, and in a lesser degree the corresponding zone of the Kalahari in the south, have an exceedingly scanty rainfall, the winds which blow over them from the ocean losing part of their moisture as they pass over the outer highlands, and becoming constantly drier owing to the heating effects of the burning soil of the interior; while the scarcity of mountain ranges in the more central parts likewise tends to prevent condensation. In the inter-tropical zone of summer precipitation, the rainfall is greatest when the sun is vertical or soon after. It is therefore greatest of all near the equator, where the sun is twice vertical, and less in the direction of both tropics. The rainfall zones are, however, somewhat deflected from a due west-to-east direction, the drier northern conditions extending southwards along the east coast, and those of the south northwards along the west. Within the equatorial zone certain areas, especially on the shores of the Gulf of Guinea and in the upper Nile basin, have an intensified rainfall, but this rarely approaches that of the rainiest regions of the world. The rainiest district in all Africa is a strip of coastland west of Mount Cameroon, where there is a mean annual rainfall of about 10000 mm as compared with a mean of 11600 mm at Cherrapunji, in Meghalaya, India.

The two distinct rainy seasons of the equatorial zone, where the sun is vertical at half-yearly intervals, become gradually merged into one in the direction of the tropics, where the sun is overhead but once. Snow falls on all the higher mountain ranges, and on the highest the climate is thoroughly Alpine.

The countries bordering the Sahara are much exposed to a very dry wind, full of fine particles of sand, blowing from the desert towards the sea. Known in Egypt as the khamsin, on the Mediterranean as the sirocco, it is called on the Guinea coast the harmattan. This wind is not invariably hot; its great dryness causes so much evaporation that cold is not infrequently the result. Similar dry winds blow from the Kalahari Desert in the south. On the eastern coast the monsoons of the Indian Ocean are regularly felt, and on the southeast hurricanes are occasionally experienced.

Health
The climate of Africa lends itself to certain environmental diseases, the most serious of which are: malaria, sleeping sickness and yellow fever. Malaria is the most deadly environmental disease in Africa. It is transmitted by a genus of mosquito (anopheles mosquito) native to Africa, and can be contracted over and over again. There is not yet a vaccine for malaria, which makes it difficult to prevent the disease from spreading in Africa. Recently, the dissemination of mosquito netting has helped lower the rate of malaria.

Yellow fever is a disease also transmitted by mosquitoes native to Africa. Unlike malaria, it cannot be contracted more than once. Like chicken pox, it is a disease that tends to be severe the later in life a person contracts the disease.

Sleeping sickness, or African trypanosomiasis, is a disease that usually affects animals, but has been known to be fatal to some humans as well. It is transmitted by the tsetse fly and is found almost exclusively in Sub-Saharan Africa. This disease has had a significant impact on African development not because of its deadly nature, like Malaria, but because it has prevented Africans from pursuing agriculture (as the sleeping sickness would kill their livestock).

title: Geography of Africa


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Approval voting

Approval voting is a single-winner electoral system in which voters mark all the candidates they support, instead of just choosing one. The candidate with the highest approval rating is elected.

Effect on elections
Research by social choice theorists Steven Brams and Dudley R. Herschbach found that approval voting would increase voter participation, prevent minor-party candidates from being spoilers, and reduce negative campaigning. Brams' research concluded that approval can be expected to elect majority-preferred candidates in practical election scenarios, avoiding the center squeeze common to ranked-choice voting and primary elections.

One study showed that approval would not have chosen the same two winners as plurality voting (Chirac and Le Pen) in the first round of the 2002 French presidential election; it instead would have chosen Chirac and Jospin as the top two candidates to proceed to the runoff.

In the actual election, Le Pen lost by an overwhelming margin in the runoff, 82.2% to 17.8%, a sign that the true top two candidates had not been found. In the approval voting survey primary, Chirac took first place with 36.7%, compared to Jospin at 32.9%. Le Pen, in that study, received 25.1% and so would not have made the cut to the second round. In the real primary election, the top three were Chirac, 19.9%, Le Pen, 16.9%, and Jospin, 16.2%. A study of various "evaluative voting" methods (approval and score voting) during the 2012 French presidential election showed that "unifying" candidates tended to do better, and polarizing candidates did worse, as compared to under plurality voting.

Operational impacts

 * Simple to tally—Approval ballots can be counted by some existing machines designed for plurality elections, as ballots are cast, so that final tallies are immediately available after the election, with relatively few if any upgrades to equipment.
 * Just one round—Approval can remove the need for multiple rounds of voting, such as a primary or a run-off, simplifying the election process.
 * Avoids Overvotes—Approval voting does not have the notion of overvotes, where voting for one more than allowed will cancel the entire opportunity to vote. In plurality elections, overvotes have to be reviewed and resolved if possible while in approval voting, no time is wasted on this activity.

Latvia
The Latvian parliament uses approval voting within open list proportional representation.

United States
In November 2020, St. Louis, Missouri, passed Proposition D with 70% voting to authorize a variant of approval (unified primary) for municipal offices. In 2021, the first mayoral election with approval voting saw Tishaura Jones and Cara Spencer move on to the general with 57% and 46% support. Lewis Reed and Andrew Jones were eliminated with 39% and 14% support, resulting in an average of 1.6 candidates supported by each voter in the 4 person race.
 * Missouri

In 2024, in the wake of the 2022 Alaska special election State Representative Ben Baker introduced legislation to ban the expansion of voting methods other than plurality statewide. It passed the House on April 15, 2024.

In 2018, Fargo, North Dakota, passed a local ballot initiative adopting approval for the city's local elections, becoming the first United States city and jurisdiction to adopt approval. Previously in 2015, a Fargo city commissioner election had suffered from six-way vote-splitting, resulting in a candidate winning with an unconvincing 22% plurality of the vote.
 * North Dakota

The first election was held June 9, 2020, selecting two city commissioners, from seven candidates on the ballot. Both winners received over 50% approval, with an average 2.3 approvals per ballot, and 62% of voters supported the change to approval in a poll. A poll by opponents of approval was conducted to test whether voters had in fact voted strategically according to the Burr dilemma. They found that 30% of voters who bullet voted did so for strategic reasons, while 57% did so because it was their sincere opinion. Fargo's second approval election took place in June 2022, for mayor and city commission. The incumbent mayor was re-elected from a field of 7 candidates, with an estimated 65% approval, with voters expressing 1.6 approvals per ballot, and the two commissioners were elected from a field of 15 candidates, with 3.1 approvals per ballot.

In 2023, the North Dakota legislature passed a bill which intended to ban approval voting. The bill was vetoed by governor Doug Burgum, citing the importance of "home rule" and allowing citizens control over their local government. The legislature attempted to overrule the veto but failed.

Use by organizations
Approval has been used in privately administered nomination contests by the Independent Party of Oregon in 2011, 2012, 2014, and 2016. Oregon is a fusion voting state, and the party has cross-nominated legislators and statewide officeholders using this method; its 2016 presidential preference primary did not identify a potential nominee due to no candidate earning more than 32% support. The party switched to using STAR voting in 2020.

It is also used in internal elections by the American Solidarity Party; the Green Parties of Texas and Ohio; the Libertarian National Committee; the Libertarian parties of Texas, Colorado,  Arizona, and New York; Alliance 90/The Greens in Germany; and the Czech and German Pirate Party.

Approval has been adopted by several societies: the Society for Social Choice and Welfare (1992), Mathematical Association of America (1986), the American Mathematical Society, the Institute of Management Sciences (1987) (now the Institute for Operations Research and the Management Sciences), the American Statistical Association (1987), and the Institute of Electrical and Electronics Engineers (1987). The IEEE board in 2002 rescinded its decision to use approval. IEEE Executive Director Daniel J. Senese stated that approval was abandoned because "few of our members were using it and it was felt that it was no longer needed."

Historical
Robert J. Weber coined the term "Approval Voting" in 1971. It was more fully published in 1978 by political scientist Steven Brams and mathematician Peter Fishburn.

Historically, several voting methods that incorporate aspects of approval have been used:

 * Approval was used for papal conclaves between 1294 and 1621, with an average of about forty cardinals engaging in repeated rounds of voting until one candidate was listed on at least two-thirds of ballots.
 * In the 13th through 18th centuries, the Republic of Venice elected the Doge of Venice using a multi-stage process that featured random selection and voting that allowed approval of multiple candidates.
 * According to Steven J. Brams, approval was used for unspecified elections in 19th century England.
 * The Secretary-General of the United Nations is elected using approval/disapproval voting in the Security Council, with the exception that permanent members of the Security Council may veto candidates.
 * Approval was used in Greek legislative elections from 1864 to 1923, after which it was replaced with party-list proportional representation.
 * Sequential proportional approval voting was used in Swedish elections in the early 20th century, prior to being replaced by party-list proportional representation.

The idea of approval was adopted by X. Hu and Lloyd Shapley in 2003 in studying authority distribution in organizations.

Overview
Approval voting allows voters to select all the candidates whom they consider to be reasonable choices.

Strategic approval differs from ranked voting (aka preferential voting) methods where voters are generally forced to reverse the preference order of two options, which if done on a larger scale can cause an unpopular candidate to win. Strategic approval, with more than two options, involves the voter changing their approval threshold. The voter decides which options to give the same rating, even if they were to have a preference order between them. This leaves a tactical concern any voter has for approving their second-favorite candidate, in the case that there are three or more candidates. Approving their second-favorite means the voter harms their favorite candidate's chance to win. Not approving their second-favorite means the voter helps the candidate they least desire to beat their second-favorite and perhaps win.

Approval technically allows for but is strategically immune to push-over and burying.

Bullet voting occurs when a voter approves only candidate "a" instead of both "a" and "b" for the reason that voting for "b" can cause "a" to lose. The voter would be satisfied with either "a" or "b" but has a moderate preference for "a". Were "b" to win, this hypothetical voter would still be satisfied. If supporters of both "a" and "b" do this, it could cause candidate "c" to win. This creates the "chicken dilemma", as supporters of "a" and "b" are playing chicken as to which will stop strategic voting first, before both of these candidates lose.

Compromising occurs when a voter approves an additional candidate who is otherwise considered unacceptable to the voter to prevent an even worse alternative from winning.

Sincere voting
Approval experts describe sincere votes as those "... that directly reflect the true preferences of a voter, i.e., that do not report preferences 'falsely. They also give a specific definition of a sincere approval vote in terms of the voter's ordinal preferences as being any vote that, if it votes for one candidate, it also votes for any more preferred candidate. This definition allows a sincere vote to treat strictly preferred candidates the same, ensuring that every voter has at least one sincere vote. The definition also allows a sincere vote to treat equally preferred candidates differently. When there are two or more candidates, every voter has at least three sincere approval votes to choose from. Two of those sincere approval votes do not distinguish between any of the candidates: vote for none of the candidates and vote for all of the candidates. When there are three or more candidates, every voter has more than one sincere approval vote that distinguishes between the candidates.

Examples
Based on the definition above, if there are four candidates, A, B, C, and D, and a voter has a strict preference order, preferring A to B to C to D, then the following are the voter's possible sincere approval votes:
 * vote for A, B, C, and D
 * vote for A, B, and C
 * vote for A and B
 * vote for A
 * vote for no candidates

If the voter instead equally prefers B and C, while A is still the most preferred candidate and D is the least preferred candidate, then all of the above votes are sincere and the following combination is also a sincere vote:
 * vote for A and C

The decision between the above ballots is equivalent to deciding an arbitrary "approval cutoff." All candidates preferred to the cutoff are approved, all candidates less preferred are not approved, and any candidates equal to the cutoff may be approved or not arbitrarily.

Sincere strategy with ordinal preferences
A sincere voter with multiple options for voting sincerely still has to choose which sincere vote to use. Voting strategy is a way to make that choice, in which case strategic approval includes sincere voting, rather than being an alternative to it. This differs from other voting systems that typically have a unique sincere vote for a voter.

When there are three or more candidates, the winner of an approval election can change, depending on which sincere votes are used. In some cases, approval can sincerely elect any one of the candidates, including a Condorcet winner and a Condorcet loser, without the voter preferences changing. To the extent that electing a Condorcet winner and not electing a Condorcet loser is considered desirable outcomes for a voting system, approval can be considered vulnerable to sincere, strategic voting. In one sense, conditions where this can happen are robust and are not isolated cases. On the other hand, the variety of possible outcomes has also been portrayed as a virtue of approval, representing the flexibility and responsiveness of approval, not just to voter ordinal preferences, but cardinal utilities as well.

Dichotomous preferences
Approval avoids the issue of multiple sincere votes in special cases when voters have dichotomous preferences. For a voter with dichotomous preferences, approval is strategyproof. When all voters have dichotomous preferences and vote the sincere, strategy-proof vote, approval is guaranteed to elect the Condorcet winner, if one exists. However, having dichotomous preferences when there are three or more candidates is not typical. It is an unlikely situation for all voters to have dichotomous preferences when there are more than a few voters.

Having dichotomous preferences means that a voter has bi-level preferences for the candidates. All of the candidates are divided into two groups such that the voter is indifferent between any two candidates in the same group and any candidate in the top-level group is preferred to any candidate in the bottom-level group. A voter that has strict preferences between three candidates—prefers A to B and B to C—does not have dichotomous preferences.

Being strategy-proof for a voter means that there is a unique way for the voter to vote that is a strategically best way to vote, regardless of how others vote. In approval, the strategy-proof vote, if it exists, is a sincere vote.

Approval threshold
Another way to deal with multiple sincere votes is to augment the ordinal preference model with an approval or acceptance threshold. An approval threshold divides all of the candidates into two sets, those the voter approves of and those the voter does not approve of. A voter can approve of more than one candidate and still prefer one approved candidate to another approved candidate. Acceptance thresholds are similar. With such a threshold, a voter simply votes for every candidate that meets or exceeds the threshold.

With threshold voting, it is still possible to not elect the Condorcet winner and instead elect the Condorcet loser when they both exist. However, according to Steven Brams, this represents a strength rather than a weakness of approval. Without providing specifics, he argues that the pragmatic judgments of voters about which candidates are acceptable should take precedence over the Condorcet criterion and other social choice criteria.

Strategy with cardinal utilities
Voting strategy under approval is guided by two competing features of approval. On the one hand, approval fails the later-no-harm criterion, so voting for a candidate can cause that candidate to win instead of a candidate more preferred by that voter. On the other hand, approval satisfies the monotonicity criterion, so not voting for a candidate can never help that candidate win, but can cause that candidate to lose to a less preferred candidate. Either way, the voter can risk getting a less preferred election winner. A voter can balance the risk-benefit trade-offs by considering the voter's cardinal utilities, particularly via the von Neumann–Morgenstern utility theorem, and the probabilities of how others vote.

A rational voter model described by Myerson and Weber specifies an approval strategy that votes for those candidates that have a positive prospective rating. This strategy is optimal in the sense that it maximizes the voter's expected utility, subject to the constraints of the model and provided the number of other voters is sufficiently large.

An optimal approval vote always votes for the most preferred candidate and not for the least preferred candidate, which is a dominant strategy. An optimal vote can require supporting one candidate and not voting for a more preferred candidate if there 4 candidates or more; however, such situations are inherently unstable, suggesting such strategy should be rare.

Other strategies are also available and coincide with the optimal strategy in special situations. For example:
 * Vote for the candidates that have above average utility. This strategy coincides with the optimal strategy if the voter thinks that all pairwise ties are equally likely.
 * Vote for any candidate that is more preferred than the expected winner and also vote for the expected winner if the expected winner is more preferred than the expected runner-up. This strategy coincides with the optimal strategy if there are three or fewer candidates or if the pivot probability for a tie between the expected winner and expected runner-up is sufficiently large compared to the other pivot probabilities. This strategy, if used by all voters, implies at equilibrium the election of the Condorcet winner whenever it exists.
 * Vote for the most preferred candidate only. This strategy coincides with the optimal strategy when the best candidate is either much better than all others (i.e. is the only one with a positive expected value.
 * If all voters are rational and cast a strategically optimal vote based on a common knowledge of how all other voters vote except for small-probability, statistically independent errors, then the winner will be the Condorcet winner, if one exists.

Strategy examples
In the example election described here, assume that the voters in each faction share the following von Neumann–Morgenstern utilities, fitted to the interval between 0 and 100. The utilities are consistent with the rankings given earlier and reflect a strong preference each faction has for choosing its city, compared to weaker preferences for other factors such as the distance to the other cities.

Using these utilities, voters choose their optimal strategic votes based on what they think the various pivot probabilities are for pairwise ties. In each of the scenarios summarized below, all voters share a common set of pivot probabilities.

In the first scenario, voters all choose their votes based on the assumption that all pairwise ties are equally likely. As a result, they vote for any candidate with an above-average utility. Most voters vote for only their first choice. Only the Knoxville faction also votes for its second choice, Chattanooga. As a result, the winner is Memphis, the Condorcet loser, with Chattanooga coming in second place. In this scenario, the winner has minority approval (more voters disapproved than approved) and all the others had even less support, reflecting the position that no choice gave an above-average utility to a majority of voters.

In the second scenario, all of the voters expect that Memphis is the likely winner, that Chattanooga is the likely runner-up, and that the pivot probability for a Memphis-Chattanooga tie is much larger than the pivot probabilities of any other pair-wise ties. As a result, each voter votes for any candidate they prefer more than the leading candidate, and also vote for the leading candidate if they prefer that candidate more than the expected runner-up. Each remaining scenario follows a similar pattern of expectations and voting strategies.

In the second scenario, there is a three-way tie for first place. This happens because the expected winner, Memphis, was the Condorcet loser and was also ranked last by any voter that did not rank it first.

Only in the last scenario does the actual winner and runner-up match the expected winner and runner-up. As a result, this can be considered a stable strategic voting scenario. In the language of game theory, this is an "equilibrium." In this scenario, the winner is also the Condorcet winner.

Dichotomous cutoff
As this voting method is cardinal rather than ordinal, it is possible to model voters in a way that does not simplify to an ordinal method. Modelling voters with a 'dichotomous cutoff' assumes a voter has an immovable approval cutoff, while having meaningful cardinal preferences. This means that rather than voting for their top 3 candidates, or all candidates above the average approval (which may result in their vote changing if one candidate drops out, resulting in a system that does not satisfy IIA), they instead vote for all candidates above a certain approval 'cutoff' that they have decided. This cutoff does not change, regardless of which and how many candidates are running, so when all available alternatives are either above or below the cutoff, the voter votes for all or none of the candidates, despite preferring some over others. This could be imagined to reflect a case where many voters become disenfranchised and apathetic if they see no candidates they approve of. In a case such as this, many voters may have an internal cutoff, and would not simply vote for their top 3, or the above average candidates, although that is not to say that it is necessarily entirely immovable.

For example, in this scenario, voters are voting for candidates with approval above 50% (bold signifies that the voters voted for the candidate):

C wins with 65% of the voters' approval, beating B with 60%, D with 40% and A with 35%

If voters' threshold for receiving a vote is that the candidate has an above average approval, or they vote for their two most approved of candidates, this is not a dichotomous cutoff, as this can change if candidates drop out. On the other hand, if voters' threshold for receiving a vote is fixed (say 50%), this is a dichotomous cutoff, and satisfies IIA as shown below:

B now wins with 60%, beating C with 55% and D with 40% With dichotomous cutoff, C still wins.

B now wins with 70%, beating C and A with 65% With dichotomous cutoff, C still wins.

Compliance with voting system criteria
Most of the mathematical criteria by which voting systems are compared were formulated for voters with ordinal preferences. In this case, approval voting requires voters to make an additional decision of where to put their approval cutoff (see examples above). Depending on how this decision is made, approval satisfies different sets of criteria.

There is no ultimate authority on which criteria should be considered, but the following are criteria that many voting theorists accept and consider desirable:
 * Unrestricted domain—A voter may have any preference ordering among the alternatives.
 * Non-dictatorship—There does not exist a single voter whose preference for the alternatives always determines the outcome regardless of other voters' preferences.
 * Pareto efficiency—If every voter prefers candidate A to all other candidates, then A must be elected.
 * Majority favorite—If there exists a majority that ranks (or rates) a single candidate higher than all other candidates, does that candidate always win?
 * Monotonicity criterion—Is it impossible to cause a winning candidate to lose by ranking that candidate higher, or to cause a losing candidate to win by ranking that candidate lower?
 * Consistency criterion—If the electorate is divided in two and a choice wins in both parts, does it always win overall?
 * Participation criterion—Is voting honestly always better than not voting at all?
 * Condorcet criterion—If a candidate beats every other candidate in pairwise comparison, does that candidate always win?
 * Condorcet loser criterion—If a candidate loses to every other candidate in pairwise comparison, does that candidate always lose?
 * Independence of irrelevant alternatives—Is the outcome the same after adding or removing non-winning candidates?
 * Independence of clones criterion—Is the outcome the same if candidates identical to existing candidates are added?
 * Reversal symmetry—If individual preferences of each voter are inverted, does the original winner never win?

title: Approval voting