## 16-cell antiprism

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### 16-cell antiprism

Given that the 16 cell is its own dual, there exists a 5 dimensional 16-cell anti-prism whose 4 dimensional 'faces' which are the two 16-cells and 32 octohedral pyramids. I'd like to know if there is any additional information on this 5 dimensional figure and whether 16-cell antiprism is its preferred name.
naraht
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### Re: 16-cell antiprism

Welcome!

It's actually the 24-choron that has 24 octahedral faces that is self-dual. The 16-choron is dual to the tesseract.

The description as prisms is indeed correct. The dual of a prism is a tegum. For a general right prism of two figures A, B, where A occupies space in a1, a2, a3 coordinates, and B is in b1, b2 space, the prism is that figure with the coordinates a1, a2, a3, b1, b2, such that a1-a3 is in A and b1, b2 is in B. For example, a hexagonal prism can be made by cutting a layer of 3-space, and then a hexagon inside this layer, or cutting a hexgaon column (ie hexagon * line), and then cutting an offcut. 'prisma' is a greek word meaning offcut (as a carpenter might make).

The duals of prisms is the tegum. You make a tegum by putting A into the space a1, a2, a3, 0, 0, and B into the space 0,0,0,b1,b2. You then cover this such as you might put cloth over poles of a tent. 'tegum' derives from a latin word, to cover.

An antiprism is a figure formed by placing a figure and its dual in parallel layers. From this, one constructs pyramids of each element of the top to the bottom. Pyramids are like tegums, except there is an extra height coordinate. When h=0, A=1 and B = 0, For other values of h (to h=1), the layers become h=x, A=1-x, B=x. You can see the tetrahedron, apart from being a triangular pyramid, is also the pyramid product of two opposite edges, each slice is a rectangle of (1-x)A, (x)B.

The dual of an antiprism is an antitegum. This is made by the intersection of cones of the duals, pointing in opposite directions. So if the axis runs from 0 to 1, the antitegum is the intersection of xA and (1-x)B. The cube is a triangle antitegum. The great diagonal is the axis, and the triangle around these vertices are the cone markers. You then see that instead of expanding forever, they are truncated by the opposite cone, so you can colour three faces around one corner red, and the other three blue. The cones (in the sense of a travelling expansion), are infinite, but the intersection is finite.
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wendy
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### Re: 16-cell antiprism

sorry, meant 24 choron for each half.

It appears that to create an anti-prism in n-space, you must start with a polytope in dimension n-1 that is its own dual, so you have the 3-space antiprisms, the anti-prisms formed from n-1 simplexes (where the anti-prism is the n-orthoplexes) and then this beast.

What is it's preferred name?
naraht
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### Re: 16-cell antiprism

Sorry meant that each half was a 24 choron. Is there a more common name than 24 choron anti-prism or 24 cell anti-prism?
naraht
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### Re: 16-cell antiprism

The 16-cell antiprism does exist in 5D, if interpreted as a 4-demicube with two constituent halves forming a tesseractic projection. Such a figure would be the resultant alternation of a tesseractic prism (which is none other than a penteract). That shape is no other than the demipenteract, which is semi-regular and uniform.

Facets are 2 hexadecachora, 8 tetrahedral antiprisms, and 16 pentachora (as tetrahedral pyramids) under alternated tesseractic prism symmetry (order 384). In contrast, its full symmetry is of order 1920, half of the 5-cube group with 10 identical hexadecachora and 16 regular pentachora for facets. Vertex figure is a rectified 5-cell.

And no, the 16-cell is not self-dual because its dual is the tesseract which has only 8 cells, which are all cubes. The 16-cell is composed of 16 identical regular tetrahedra.

The usual definition of antiprisms as alternated prisms directly contrasts with Wendy's definition. Antiprisms are alternations of prisms with even faces, with the alternated caps as bases. For example, the nonuniform pyritohedral icosahedral antiprism (pyikap) is formed from an alternation of a truncated octahedral prism with the two icosahedra (in pyritohedral symmetry) as bases. Polytopes composed of a specific base and its dual base are not usually considered antiprisms, but rather bidual segmentotopes.

Wendy's definition of a 24-cell is actually a bidual degenerate tes-hex segmentoteron, composed of a tesseract base and a 16-cell base. The lateral facets are supposed to be cubic pyramids, but since in the 24-cell the square pyramid cells merge (coinciding with equal lengths), there is no need for an additional dimension since the height between the bases are already 0.

Welcome to the forum and have a nice day!
Mercurial, the Spectre
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### Re: 16-cell antiprism

First of all, the 3D antiprisms allow for both interpretations, either as having identical, but gyrated bases, or as having bases, which are each others duals. Both interpretations extend to higher dimensions, but they would differ then. Mercurial's 16-cell "antiprism" uses the first reading, while Wendy's 24-cell "antiprism" uses the second.

In order not to mix those up it ought be desirable to distinguish those namings then. This is also the aim of Mercurial's provided namings. Sadly he has not read my original paper on Convex Segmentochora carefully. Else he would know that the word antiprism there indeed already had been used in Wendy's reading.

The purpose for this choice also is outlined on my website in detail: Mercurial's Interpretation of antiprism as having gyrated bases amounts in a snub operation. Sadly that operation (considered as uniform representations thereof) within higher dimensions often runs into problems with the available degrees of freedom, thus becoming there rather scarce. Therefore it is a better idea to use that term in the meanwhile settled (second) reading.

The other (first) reading (of Mercurial) there has been called a gyroprism. Thus Mercurial's example, the hemipenteract, should rather be attributed the 16-cell gyroprism.

By the way, there is, none the less, a true 16-cell antiprism (even in the second reading). The dual of the 16-cell is the 8-cell or tesseract, therefor it likewise could be considered a tesseract antiprism: Just stack these two within 5D parallely atop each other and you'll get tessap with 1 16-cell, 16 5-cells (used as tetrahedral pyramids), further 32 5-cells (used as line atop fully ortho triangle), 24 squippipies (line atop fully ortho square), 8 cube pyramids, and 1 tesseract.

--- rk
Klitzing
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### Re: 16-cell antiprism

naraht wrote:sorry, meant 24 choron for each half.

It appears that to create an anti-prism in n-space, you must start with a polytope in dimension n-1 that is its own dual, so you have the 3-space antiprisms, the anti-prisms formed from n-1 simplexes (where the anti-prism is the n-orthoplexes) and then this beast.

What is it's preferred name?

Sorry if I find it confusing. We are into a debate on what antiprisms are.

The 24-cell antiprism is already mentioned by Klitzing in his bendwavy site and is isogonal. It is called an icositetrachoric antiprism (icoap), derived from the tetradisphenoidal 288-cell prism via identifying a compound of two 24-cells from each base (the tetradisphenoidal 288-cell is the dual of the bitruncated 24-cell and contains the vertices of two congurent 24-cells) and deleting alternate sets, making sure that the result is not centrally symmetric. Facets are 2 icositetrachora, 48 octahedral pyramids, and 192 pentachora. It can be made scaliform but not uniform due to the octahedral pyramid facets. It has a symmetry order of 2304. Its projection into 4D is the tetradisphenoidal 288-cell.

Klitzing says that antiprisms are of Wendy's definition, when they could be more properly called bidual segmentotopes. The definition of a prism refers to two congruent bases, hereafter the subcategories such as antiprisms and gyroprisms should mean the same, having two congruent bases rotated at some axis relative to each other. Notice how it is called an antiprism because it contains two congruent bases, and Klitzing's said 16-cell antiprism would be termed as a bidual tes-hex segmentoteron. His idea relies on the generalization of duality, since then they cannot be derived from alternations anymore. Also, the bias of uniformity means that he would rather choose the second definition, when it is actually a geometrically accepted fact and has nothing to do with the preference of his kind of polytopes. When I read his paper and his page, I noticed that he was confused with the use of gyroprisms with another of the 6D variety, and he provided the mix-up in the first place.

We should restrict the use of prism generalizations as any polytope formed from two sets of congruent vertices lying at two parallel hyperplanes. Reading the paper only contributes of the mix-up between antiprisms and their intended definition. I feel that these bidual segmentotopes are not even antiprisms, but rather an ordinary segmentotope. Notice how Tomruen uses the term "antiprism" in the alternations of truncated prisms. A 2-3 duoantiprism is even related to an antiprism because of its name, similar to how a duoprism is related to a prism.

And, finally, by considering the definition of isogonality, antiprisms are usually isogonal in their highest-symmetry form because of vertex congruence between the bases.

We should agree, and let's settle things down.
Mercurial, the Spectre.
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### Re: 16-cell antiprism

The anti-prism is a specific and special example of lace prisms. The special features is that sections of various transforms to regular figures, form slices of the anti-prism and its dual (anti-tegum). In any case, there is a specific construction of lace prisms and their duals (lace tegums), which extend far beyond Richard's segmentotopes, in the way that prism-products extend well past ordinary slab-prisms.

The construction by removing alternate vertices of figures, in 3D polygonal prisms, will give anti-prisms, but this is not general. In four dimensions, you don't really get the same result . The example given of xo3oo3oxAoo&#xt, is the general half-cube in every dimension, is a semiated (or alternately-diminished) cube. One can construct figures where in place of alternate, remove five out of six vertices, or dare we say, 'decimated'. Such a figure of 216 vertices has been my worry in eight dimensions.

It does not do well to suppose that different constructions that yield the same figure in 3D or 4D, will do so in every dimension. There are four relatively important figures, say A, B, C, D, where in 2D, we have A=B=square, C=D = hexagon. In 3D, we get A=C=cuboctahedron, B=D=rhombic dodecahedron. In 4D, we get A=B=24choron, C is the runcinated simplex, and D is the strombiated simplex. After 4D, A=rectified orthotope, B=apiculated polyact. The four figures all have interesting features, but to suppose that A=B because they are equal in 2D and 4D is to fall into the trap of 3D is every dimension.

An example of this is the rhombus. In 2D, it serves both the tegum or shell of lines crossing at right angles, and as an oblique parallelopied. But in higher dimensions, including 3D, these are distinct figures (octahedron vs cube or 16ch vs tesseract). The rhombohedron is a figure according to the oblique cube, while the rhombic dodecahedron / tricontahedron, are due to the tegum-product of the axies. In 4D, the dual of a rectified polychoron has not rhombohedra for faces, but the line-tegums. These become surtegmated figures, where the surface tegums are the tegum-product of the corresponding elements of dual polytopes.
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### Re: 16-cell antiprism

Mercurial, the Spectre wrote:We are into a debate on what antiprisms are.

Indeed!

Once again, Mercurial is refering more to the second part of the term "antiprism", so suggestively asking for 2 identical bases, whereas I refer to the first part of that term "antiprism", which clearly refers to duality. The prism part here then just refers to the stacking operation. (Just as Wendy's term of "laceprisms" herewith only refers to that stacking operation.)

His argument wrt. 4D duoantiprisms is not valid, as there again we are back to 2D polygons, where the gyration and the duality come out to be the same.

Also he was reading my paragraph on gyroprisms wrong. Gyroprisms indeed are stacks - or more generally simplices of mutually gyrated (or obtionally reflected for n=2). Thus the refered 6D candidate is clearly there called a "gyrotrigonism", e.g. hexgyt = tedjak = xoo3ooo3oxo *b3oox&#x , whereas hin indeed is a hexgyp = xo3oo3ox *b3oo&#x.

Infact there is no predefined "geometrical fact" which relates the term "antiprism" to snubs (alternations). It just happens within 3D to fall together. But this is just because of gyroprisms are related to alternations (generally).

Btw. Mercurial's description of the icoap as a true alternation (this time derived from a prism of some Catalan hypersolid) indeed is correct. But obviously he is cheating here, as he applies snubbing only onto a dual of the appropriate compound.

I also agree in the isogonality, but wrt. gyroprisms (and gyrosimplices). Antiprisms (within my usage of the term), i.e. stacks of dual polytopes, only become isogonal if the base happens to be selfdual (like all 2D polygons). So e.g. hex = tet || dual tet is isogonal, but octap = oct || cube and ikap = ike || doe are not.

--- rk
Klitzing
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### Re: 16-cell antiprism

Kepler invented 'antiprism'. I defered to him.

Ultimately, antiprism and antitegum are much more than strata figures. You can have figures that are antiprisms in 64 different ways, if you want to. The tegum-product of seven antiprisms is an antiprism of the pyramid product of the each of any of the seven bases, and since the antiprism is the same as that of the dual, of 64 different figures.
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wendy
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### Re: 16-cell antiprism

Klitzing wrote:
Mercurial, the Spectre wrote:We are into a debate on what antiprisms are.

Indeed!

Once again, Mercurial is refering more to the second part of the term "antiprism", so suggestively asking for 2 identical bases, whereas I refer to the first part of that term "antiprism", which clearly refers to duality. The prism part here then just refers to the stacking operation. (Just as Wendy's term of "laceprisms" herewith only refers to that stacking operation.)

His argument wrt. 4D duoantiprisms is not valid, as there again we are back to 2D polygons, where the gyration and the duality come out to be the same.

Also he was reading my paragraph on gyroprisms wrong. Gyroprisms indeed are stacks - or more generally simplices of mutually gyrated (or obtionally reflected for n=2). Thus the refered 6D candidate is clearly there called a "gyrotrigonism", e.g. hexgyt = tedjak = xoo3ooo3oxo *b3oox&#x , whereas hin indeed is a hexgyp = xo3oo3ox *b3oo&#x.

Infact there is no predefined "geometrical fact" which relates the term "antiprism" to snubs (alternations). It just happens within 3D to fall together. But this is just because of gyroprisms are related to alternations (generally).

Btw. Mercurial's description of the icoap as a true alternation (this time derived from a prism of some Catalan hypersolid) indeed is correct. But obviously he is cheating here, as he applies snubbing only onto a dual of the appropriate compound.

I also agree in the isogonality, but wrt. gyroprisms (and gyrosimplices). Antiprisms (within my usage of the term), i.e. stacks of dual polytopes, only become isogonal if the base happens to be selfdual (like all 2D polygons). So e.g. hex = tet || dual tet is isogonal, but octap = oct || cube and ikap = ike || doe are not.

--- rk

Hmm... let me explain.

First, the term "anti" means opposite, and your comparison with duals seems off-putting. Nobody calls a regular octahedron as an anticube, because that term is taken for the square antiprism. The prism part is okay, because traditionally prisms have congruent bases. If antiprisms mean dual-stacking, then we are referring to shapes such as cube||oct, but the cube and the octahedron are simply duals, not antialigned congruent shapes. A fitting description for an octahedral antiprism would be the alternation of a hexagonal-prismatic prism, in which the two hexagonal prism bases turn into octahedra directed anti or opposite to each other so that its lateral sides form trigonal antiprisms, so its status as an antiprism makes sense. To be clear, an antiprism is similar to the prism except that the bases (which are identical) are antialigned to each other.

About the 4D duoantiprisms, they are generalizations of antiprisms since the bases alternate with each other, and they are extensions of 3D antiprisms. Strictly they aren't antiprisms, but form a comparison using identical bases. This lends credence to the fact that antiprisms are described by bases that are identical. Only the 2-n duoantiprisms are true antiprisms.

And for the gyroprism part, I'm sorry. I agree with the 6D term, but please, antiprisms are just a subset of these.

From what I've observed, let's look at the base. If that base has a double symmetry extension while preserving all symmetry on that shape, such as the 24-cell's relationship with contic symmetry, then an antiprism can be formed. It implies that these bases can form double-symmetric 2-compounds, and if these are drawn away into hyperspace, their hulls will form antiprisms. In 2D, the regular n-gons can be 2-compounded to produce a regular 2n-gon, and they can be moved into the Z-axis to form an antiprism. Same goes with the 24-cell, in which a compound of two will produce the dual bitruncated 24-cell, having double the symmetry. It can then be moved into the 5D axis to form the 24-cell antiprism. The cube has one, those of a square prism. Compounding the 2 so that it produces an octagonal prism, and then moving into the W-axis, we get the square-antiprism prism, or in my own opinion, the cubic antiprism. The octahedron has one, and that is with the trigonal antiprism, and its 2-compound forms the hexagonal prism, we get a 2-3 duoantiprism or the octahedral antiprism (which I mentioned above) if we extrude the bases to the 4th dimension. Then the double-symmetry result can form a prism, of which the chosen antiprism represents a form of "partial" snubbing (i.e. not a true alternation). They are related due to their symmetries.

My argument about the icoap is covered above. And there is no cheating involved; there is a particular subsymmetrical faceting within the tetradisphenoidal 288-cell prism that corresponds to the shape.

Feel free to suggest on solutions, maybe I'm mistaken. You know, we're helping the newcomer.
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### Re: 16-cell antiprism

The main problem here are not the geometric figures as such. There we both agree, even within their geometric constructions.

Rather it is the attribution with terms, which are purely extrapolations of well-known terms for lower-dimensional solids. For, these specific solids do allow for multiple building principles. Most of these can be uplifted, but generally their according rersults generally do differ. Therefore you might call one higher dimensional form an "antiprism" with the very same right as I do with an other. Sorry, there cannot be any true or false. This is just is an issue of extrapolation of terms, which on either side wrongly is being assumed to be understood in just a single way.

The best thing would be to describe the building principle from scratch, i.e. without taking refuge to potentially ambiguous terms. For instance by laceprism notations if applicable.

N(-gon)-antiprism = xoNox&#x = xNo || oNx
and thus I use alike
hex = tet-antiprism = xo3oo3ox&#x = x3o3o || o3o3x
octap = xo3oo4ox&#x = x3o4o || o3o4x
icoap = xo3oo4oo3ox&#x = x3o4o3o || o3o4o3x

N(-gon)-gyroprism = xNo || gyro xNo = xNo || oNx
and thus I use alike
hin = x3o3o *b3o3o = hex-gyroprism = x3o3o *b3o || gyro x3o3o *b3o = x3o3o *b3o || o3o3x *b3o

alternated 2N(-gon)-prism = snub(x2No || x2No) = s2s2No = s2sNs = snub(xNx || xNx) = xNo || oNx
hin = alternated penteract = s4o3o3o3o = snub(x4o3o3o3o) = snub(x4o3o3o || x4o3o3o) = x3o3o *b3o || o3o3x *b3o

Etc.

--- rk
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### Re: 16-cell antiprism

While the 16-cell can be derived from the teseract by alternating vertices, there is also a derivation of the 16-cell by way of thirds of the 24-choron. The three 16-cells then sit together as

xoo3ooo3oxoAoox

This can be rended to a six-dimensional lace prism by the addition of &#x at the end, which would cause a triangle of these "16-cell antiprisms", each pair deriving a different penteract they come from.

Prisma is derived from a greek word 'to cut'. In three dimensions, the only prisms derived are those where a line is one of the factors. But in four dimensions, one can derive prisms where there is no opposite face. The bi-triangular pyramid, or triangle-duoprism, sitting on one face, the upmost surtope is not a face, but the triangle margin between two faces. Not withstanding, this prism translates to on the w,x space, a triangle, and on the y,z space, a triangle.

While prisma derives from 'offcut', such as one might strike from a rod of a given section, a different way of making prisms is to roll a layer of dough, and cut shapes using a pastry or cookie cutter. The first cut is then to roll the layer of pastry, the second is to use the cookie cutter to make the shapes.

The trouble with science, is that the variation of word use from art to art, causes many of the confusions to students, and one gets obnoxious comments on the likes of Wikipedia, where people wrongly suppose that weight is a force in 'weights and measures' (one art), where in the matter of mechanics it is a force. One only has to look at the talk-pages of things like pound, troy pound, etc to see this.

To this end, it is best to study what the word in common use means, and to make the central meaning of the word in the current art take the same. For example, physics does have a meaning corresponding to 'weight' in common use, but because they misappropriated weight to something else, they had to invent a new word. So we have 'mass', which is elsewhere something that is measured with a ruler (Mass und Gewichte).

Reading the old geometries, such as that of D M Y Sommerville or H S M Coxeter, does not help here. Coxeter uses 'cell' in the sense of face, and sometimes as a three-dimensional element. Sommerville uses 'hedroid' for a choron.

The trick we use is to suppose that the words in common use are relative to solid space, so that a plane is represented by one equal sign (ie z=... ), and that a margin is by two equal signs, eg y=., z=. New names are then created for the meanings of space of specific dimension, this derived by a back-forming of the word 'polyhedron' = poly (many + closure) + hedr (2D) + on (patch). Using various different elements it is possible to invent a word to describe E6, as a horo (E) + ect (6d) + ix (fabric).

The products are according to construction, not outcome. A prism is supposed to be cut from shapes of its bases in orthogonal axies. So a pentagon prism can be struck from a pentagon-bar (ie x,y,z -> pentagon, z -> pentagon, height), or by pressing a pentagonal cutter against a slab, ie x,y,z -. x,y, height -> pentagon, height. There are something like five known products, all of which reduces one or more properties of the bases, to algebraic forms that multiply to the property of the result.

With prisms, the dual is tegum. Cut and cover, so to speak. For all of the various uses of prism, there is a matching construction to the dual, which is a tegum. The antiprism, as constructed by the draught of duals, provides us with interesting insights of the figures described. The tegum-product of several anti-prisms is itself an antiprism, being the pyramid product of any base of each of the elements.

The terminology of ordinary geometry does not scale well. A plane in 3D, is ++=, that is a 2-dimensional thing given by a single equal-sign (z=0). But the reality is, that in four dimensions, there is no such thing. It requires four items, ie +++= or ++==. One can scale it either way. But the students are coming into the class with in mind, the plane is a dividing space, ie z=0, gives ~=. The ~ is then filled with as many + as needed. But to invent a new word for this meaning is to create some sort of confusion in the reader that a plane qua ++== can be woven as a cloth, but you can't weave +++=.

And this is why i spend a very large measure of time restructuring the words to match the expectations of the reader, rather than 'first impressions' of the author. It's not a popular path, but I find it quite necessary, because too much exotic happens up there.
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