## Johnsonian Polytopes

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

### Re: Johnsonian Polytopes

I've created a wiki page for this: CRF polychora discovery project

I've also coined the term "extratruncate" to refer to the CD string xoo...oox, i.e. the 3D cantellate and 4D runcinate. See the Truncation page for etymology.

Keiji

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### Re: Johnsonian Polytopes

quickfur wrote:I just thought of another polychoron with square antiprisms: this one has 4 square antiprisms, so it's non-trivial. Basically, take 4 of these antiprisms and join them together by their square faces, then fold them up to form a 4-membered ring in 4D. If I'm not wrong, you should be able to fill in the gaps with tetrahedra and square pyramids. I think all edge lengths should be equal, so this should be a CRF polychoron too. I'm still trying to figure out how many tetrahedra/pyramids are needed to close up the shape.

It looks like alternated 8,4-duoprism. You'll need 24 tetrahedra to fill gaps, but I'm not sure that they will be regular. To make uniform antiprism you have to take high 8-prisms as cells, so instead of cubes you'll get non-regular square prisms. And resulting tetrahedra will have different edge lengths But may be there is more that one way to fill gaps? I doubt it - we have only one convex hull of vertices.

Attached is the file with model of "ortho" for my viewer...
Attachments
ortho.txt
"ortho" model
Mrrl
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### Re: Johnsonian Polytopes

Looks like there is complete family of skew 3,n-duoprisms: you make combine two n-antiprisms with 1 n-prism, n tetrahedra and n square pyramids and get CRF body. Contructions based on 4,n rings give nothing new: if we take 2 prisms and 2 antiprsms, we get antiprismatic prism that is uniform, and for 4 antiprisms I guess that resulting body should be either uniform or non-CRF. But we don't have such uniform polychora in the list, so there is no new CRF in that side.

But what is skew 3,3? It has one triangular prism, 3 square pyramids attached to its sides, 3 tetrahedra between them and the rest is filled by two octahedra. 6 cells, 9 vertices. Does it really exist?
Last edited by Mrrl on Tue Nov 22, 2011 12:30 am, edited 1 time in total.
Mrrl
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### Re: Johnsonian Polytopes

Mrrl wrote:Coordinates of "ortho" are (±(1+sqrt(2)),±1,0,0), (±1,±(1+sqrt(2)),0,0), (±1,±1,1,±1). For "gyro" it's more difficult, because height of 4-antiprism is a difficult value itself

OK, here's a render of the ortho variant:

I freely reoriented in order to get the best view. Because this is a relatively simple polytope, I turned off visibility clipping, so you should be able to locate all the cells here. The 4 tetrahedra, 4 triangular prisms, and cube should be immediately obvious. There is actually an octagonal face cutting through the middle of this thing, which isn't easy to see (maybe i'll re-render this with a slightly different color for the octagon); the two resulting halves then are the two square cupola.

OK, here's a slightly more enhanced render in an attempt to show the octagon:

It turns out that the octagon covers the entire 2D area of the 3D->2D projection, so changing the color of the octagon didn't help make it stand out. So instead, I colored the cube red so that you can see where the octagon intersects with it. And I made the cyan for the other ridges more opaque so that its presence is more easily seen.
Last edited by quickfur on Tue Nov 22, 2011 5:33 pm, edited 1 time in total.
quickfur
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### Re: Johnsonian Polytopes

Thanks! To begin with it didn't look like the same shape, and I was about to moan about you missing out edges, but then I realized it did have all the facets after all. It just shows how much cleaner things look when rendered properly rather than just sketched in Inkscape

Keiji

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### Re: Johnsonian Polytopes

Mrrl wrote:
quickfur wrote:I just thought of another polychoron with square antiprisms: this one has 4 square antiprisms, so it's non-trivial. Basically, take 4 of these antiprisms and join them together by their square faces, then fold them up to form a 4-membered ring in 4D. If I'm not wrong, you should be able to fill in the gaps with tetrahedra and square pyramids. I think all edge lengths should be equal, so this should be a CRF polychoron too. I'm still trying to figure out how many tetrahedra/pyramids are needed to close up the shape.

It looks like alternated 8,4-duoprism. You'll need 24 tetrahedra to fill gaps, but I'm not sure that they will be regular. To make uniform antiprism you have to take high 8-prisms as cells, so instead of cubes you'll get non-regular square prisms. And resulting tetrahedra will have different edge lengths But may be there is more that one way to fill gaps? I doubt it - we have only one convex hull of vertices.

I thought it was the alternated 8,4-duoprism too, but then I'm no longer sure about that. It may be that the ring of square antiprisms doesn't make 90° internal angles, so the antiprisms form two pairs with different angles between them. So the coordinates would be different from the alternated 8,4-duoprism. There's still a chance it may be CRF. I'm still trying to work out the details.

Mrrl wrote:Looks like there is complete family of skew 3,n-duoprisms: you make combine two n-antiprisms with 1 n-prism, n tetrahedra and n square pyramids and get CRF body. Contructions based on 4,n rings give nothing new: if we take 2 prisms and 2 antiprsms, we get antiprismatic prism that is uniform, and for 4 antiprisms I guess that resulting body should be either uniform or non-CRF. But we don't have such uniform polychora in the list, so there is no new CRF in that side.

I think square antiprisms may be a special case, though, because the ring closes up fast enough that it may be possible to fill in the remaining gaps with johnson polyhedra. With larger rings, the chances of this becomes more and more unlikely because the resulting edge topology becomes too complicated and may not be able to fit regular polygons anymore.

But what is skew 3,3? It has one triangular prism, 3 square pyramids attached to its sides, 3 tetrahedra between them and the rest is filled by two octahedra. 6 cells, 9 vertices. Does it really exist?

I think so! Your cell list seems correct, and from what I can tell, it should be possible to make all edge lengths equal. But there's a danger that the square pyramids may be trapezoidal pyramids though. So this is not 100% confirmed yet.
quickfur
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### Re: Johnsonian Polytopes

quickfur wrote:
Mrrl wrote:[...]
But what is skew 3,3? It has one triangular prism, 3 square pyramids attached to its sides, 3 tetrahedra between them and the rest is filled by two octahedra. 6 cells, 9 vertices. Does it really exist?

I think so! Your cell list seems correct, and from what I can tell, it should be possible to make all edge lengths equal. But there's a danger that the square pyramids may be trapezoidal pyramids though. So this is not 100% confirmed yet.

OK, a quick (informal) check with projection diagrams (buahaha) confirms that the square pyramids are CRF. So there are 2 octahedra, 1 triangular prism, 3 tetrahedra and 3 square pyramids.

And the construction is general; the shapes of the lateral cells are independent of the degree of the degree of the antiprisms and prisms (only the number is different). So there is such a family: for all n>=3, the corresponding member of this family has a 3-membered ring made with two n-gonal antiprisms, 1 n-gonal prism, n square pyramids, and n tetrahedra. In other words, this is an infinite family of CRF polychora. Would it be the first non-uniform infinite family discovered? Way to go, Andrey!

The existence of this family can be confirmed mathematically by noting that if we attach the two antiprisms together first, then the height of the prism controls the width of two parallel edges in the square pyramids, but since this height is free to vary, it can always be set to the edge length. So you always get equal edge lengths, and so the prism has square faces. So now you have this ring of faces on the pyramids: square, triangle, triangle, and so the other two triangles must also be equilateral. Likewise, the tetrahedra must be regular because all edge lengths are equal. So the existence of this family is proven.
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### Re: Johnsonian Polytopes

You should to note also that height of antiprism is more than half of prism's height, so the triangle really exists
Mrrl
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### Re: Johnsonian Polytopes

quickfur wrote:I thought it was the alternated 8,4-duoprism too, but then I'm no longer sure about that. It may be that the ring of square antiprisms doesn't make 90° internal angles, so the antiprisms form two pairs with different angles between them. So the coordinates would be different from the alternated 8,4-duoprism. There's still a chance it may be CRF. I'm still trying to work out the details.

What if you take two n,3 skew duoprisms and glue the together by the prismatic cells? Is there a chance to get convex body? Also we may try to glue them by antiprismatic cells (and get two new bodies).
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### Re: Johnsonian Polytopes

Mrrl wrote:You should to note also that height of antiprism is more than half of prism's height, so the triangle really exists

You're right, otherwise the antiprisms have to be stretched and the polychoron will no longer be CRF.

Well, if we fix edge length to L, then an n-gonal prism always has height L. As n approaches infinity, the band of triangles of an n-gonal antiprism approaches the horizontal tiling of equilateral triangles, whose height is the height of an equilateral triangle, L*sqrt(3)/2. Since sqrt(3)/2 > 1/2, the height of an n-gonal antiprism is never shorter than half the height of the corresponding n-gonal prism, so we're safe: all members of this family exist.
quickfur
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### Re: Johnsonian Polytopes

Small comment to the discussion in Wiki: you can build a CRF pyramid over CRF polyhedron only if it is inscribed in the sphere - otherwise legths of triangles may be different. The same is about bipyramids: vertices of the base lay on the intersection of two hypespheres that is 2D sphere. So the base polyhedron should be inscribed in the sphere again. And the radius of sphere should be less than edge length.
Mrrl
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### Re: Johnsonian Polytopes

Coordinates of "gyro":

octagon: (±(1+sqrt(2)),±1,0,0), (±1,±(1+sqrt(2)),0,0)
first square: (±1,±1,S,H)
second square: (±sqrt(2),0,S,-H), (0,±sqrt(2),S,-H)

here H=sqrt(sqrt(1/2)), S=sqrt(2-sqrt(1/2)).
Mrrl
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### Re: Johnsonian Polytopes

Mrrl wrote:Coordinates of "gyro":

octagon: (±(1+sqrt(2)),±1,0,0), (±1,±(1+sqrt(2)),0,0)
first square: (±1,±1,S,H)
second square: (±sqrt(2),0,S,-H), (0,±sqrt(2),S,-H)

here H=sqrt(sqrt(1/2)), S=sqrt(2-sqrt(1/2)).

Are you sure your H value is correct? I generated the convex hull for these coordinates but got inconsistent edge lengths (2 and 3.46).

On another note, the ring of 4 square antiprisms is not a CRF polychoron. Just confirmed that it has tetrahedra with different edge lengths.
quickfur
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### Re: Johnsonian Polytopes

i'm not sure in anything. Do you know what are the points that gave you this distance? And 3.46 is actually 2*sqrt(3)=3.4641016?
Mrrl
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### Re: Johnsonian Polytopes

Actually, nevermind. Your coordinates are correct. I don't know how I ended up with the wrong coordinates; I think I probably copy-and-pasted the wrong value from the calculator I used to evaluate H and S, so it screwed up the result. It's working now. Sorry for the false alarm!
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### Re: Johnsonian Polytopes

Alright, after fixing the wrong coordinates that I don't know how it got into my polytope definition, I verified that all edge lengths are equal, and here's a nice projection of the result:

Again, I turned off visibility clipping because this is a relatively simple polytope. There is an octagonal face cutting through the middle of the image, like the previous ortho version, where the two cupola meet. So anyway. Have at it.

Side note: it's not so easy to tell from the 3D viewpoint, but the projection image is quite narrow; the radius of the octagon is significantly larger than the height of the square antiprism. So this particular projection image is almost like a lens shape.

Thanks, Mrrl, for the coordinates!
Last edited by quickfur on Tue Nov 22, 2011 5:34 pm, edited 1 time in total.
quickfur
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### Re: Johnsonian Polytopes

BTW, I just got an inspiration about naming Keiji's polychora. I shall name them the cubic ortho-square-cupola tacotope and the square antiprismic gyro-square-cupola tacotope.

You see, a taco is basically a bunch of meat and veggies sandwiched by a circular tortilla skin folded in half. So the 4D analogue of a taco is a bunch of meat and veggies sanwiched by a spherical skin folded in half. The two cupolae, resemble halves of a 4D taco skin (if you unfold them back into 3D, the approximate a sphere), and they are joined at their octagonal bases so they are basically 4D taco shells. The cube and the square antiprism are the "meat" of the taco, and the triangular prisms, tetrahedra, and square pyramids are the veggies, and all of these are sandwiched between the two halves of the taco shell. So these polychora are tacotopes.
quickfur
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### Re: Johnsonian Polytopes

And we can generalize this name to Mrrl's family of antiprismic CRF polychora too: they are just tacotopes with exotic-shaped skins, but they are essentially two halves of a shell with stuff sandwiched in between. So that family of tacotopes would be called the n-prismic n-antiprism tacotopes.

The first modifier is an adjectival form, because the meat of the taco is what gives it its name, for example, a taco containing fish is called a fish taco, a taco containing beef is called a beef taco, so the polyhedron of the 3-membered ring that isn't part of the skin gets to be the "meat" in adjectival form. The second modifier is in nominative form because it describes what kind of shell the taco has. For example, a taco with a hard shell is called a hard taco, so a taco made of two n-antiprisms is called an n-antiprism tacotope.
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### Re: Johnsonian Polytopes

Seems that diminished icositetrachora ARE part of augmented tesseracts, EXCEPT:

Icositetrachoron has 24 vertices, while tesseract only has 8 cells. Augmented tesseracts correspond to icositetrachora with vertices cut off from a specific 8-vertex subset. But it should be possible to cut off two or more vertices from DIFFERENT subsets, if they are far enough from each other - and those shapes wouldn't correspond to any augmented tesseract.

I think that the search for Johnsonian polychora needs to be two-fold:

1. Search for the primitive polychora, i.e. those that can't be diminished.
2. Put them together in various convex ways (bearing in mind that, unlike 3D case, here it's possible for two cells to be cohyperplanar if it means they will create a larger cell.

I'd suggest starting by enumerating the possible configurations of vertices and edges.

Let's look at possible dihedral angles first (all numbers in degrees). I assume that heptagonal, nonagonal and 11+-gonal prisms and antiprisms can't form anything more than duoprisms and antiprismatic prisms (although... some of those might be possible to augment?), so I omit those. Angles were taken from Great Stella.

So far it contains dihedral angles up to gyroelongated square dipyramid, feel free to extend

31,7175 - 4-10 angle of pentagonal cupola

37.3774 - 3-5 angle of pentagonal pyramid
3-10 angle of pentagonal cupola
Relations: 138.19 - 100.812, 127.377 - 90, half of 74.7547

45 - 4-8 angle of square cupola
Relations: half of 90, third of 135, 135 - 90

54.7356 - 3-4 angle of square pyramid
4-6 angle of triangular cupola
3-8 angle of square cupola
Relations: half of 109.471; 125.264 - 70.5288, 144.736 - 90, 158.572 - 103.836

60 - 4-4 angle of triangular prism, 4-4 angle of elongated triangular pyramid, 4-4 angle of elongated triangular dipyramid
Relations: Complementary to 120, half of 120

63.4349 - 5-10 angle of pentagonal rotunda
Relations: 142.623 - 79.1877

70.5288 - the dihedral angle of tetrahedron, 3-3 angle of elongated triangular pyramid, axial angle of triangular dipyramid, 3-3 angle of elongated triangular dipyramid
6-6 angle of truncated tetrahedron
3-6 angle of triangular cupola
Relations: Complementary to 109.471; 125.264 - 54.7356, 160.529 - 90, half of 141.058

74.7547 - equatorial angle of pentagonal dipyramid
Relations: double of 37.3774

79.1877 - 3-10 angle of pentagonal rotunda
Relations: 142.623 - 63.4349

90 - the dihedral angle of cube, 4-4 angle of elongated square pyramid, 4-4 angle of elongated square dipyramid
8-8 angle of truncated cube
3-4 angle of triangular prism, base 3-4 angle of elongated triangular pyramid
4-5 angle of pentagonal prism, 4-5 angle of elongated pentagonal pyramid
4-6 angle of hexagonal prism
4-8 angle of octagonal prism
4-10 angle of decagonal prism
Relations: double of 45, 135 - 45, 144.736 - 54.7356, 160.529 - 70.5288, 127.337 - 37.3774

95.2466 - 3-10 angle of decagonal antiprism

96.5945 - 3-8 angle of octagonal antiprism

98.8994 - 3-6 angle of hexagonal antiprism

100.812 - 3-5 angle of pentagonal antiprism, 3-5 angle of gyroelongated pentagonal pyramid
Relations: 138.19 - 37.3774

103.846 - 3-4 angle of square antiprism, 3-4 angle of gyroelongated square pyramid
Relations: 158.572 - 54.7356

108 - 4-4 angle of pentagonal prism, 4-4 angle of elongated pentagonal pyramid

109.471 - dihedral angle of octahedron, 3-3 angle of square pyramid, 3-3 angle of elongated square pyramid, pyramid 3-3 angle of gyroelongated square pyramid, 3-3 angle of elongated square dipyramid, pyramidal angle of gyroelongated square dipyramid
3-6 angle of truncated tetrahedron
6-6 angle of truncated octahedron
Relations: Complementary to 70.5288, double of 54.7356

116.565 - dihedral angle of dodecahedron, 10-10 angle of truncated dodecahedron

120 - 4-4 angle of hexagonal prism
Relations: Complementary to 60, double of 60

125.264 - dihedral angle of cuboctahedron, 3-4 angle of triangular cupola
4-6 angle of truncated octahedron
3-8 angle of truncated cube
6-8 angle of truncated cuboctahedron
Relations: 54.7356 + 70.5288

127.377 - 3-4 angle of elongated pentagonal pyramid
Relations: 37.3774 + 90

127.552 - 3-3 angle of square antiprism, equatorial 3-3 angle of gyroelongated square pyramid, antiprismatic angle of gyroelongated square dipyramid

135 - 4-4 angle of rhombicuboctahedron, 4-4 angle of octagonal prism, 4-4 angle of square cupola
4-8 angle of truncated cuboctahedron
Relations: triple of 45, 45 + 90

138.19 - dihedral angle of icosahedron, 3-3 angle of pentagonal antiprism, 3-3 angle of pentagonal pyramid, 3-3 angle of elongated pentagonal pyramid, 3-3 angle of gyroelongated pentagonal pyramid, axial angle of pentagonal dipyramid
6-6 angle of truncated icosahedron
Relations: 37.3774 + 100.812

141,058 - equatorial angle of triangular dipyramid
Relations: double of 70,5288

142.623 - dihedral angle of icosidodecahedron, 3-5 angle of pentagonal rotunda
5-6 angle of truncated icosahedron
3-10 angle of truncated dodecahedron
6-10 angle of truncated icosidodecahedron
Relations: 63.4349 + 79.1877

142.983 - 3-4 angle of snub cube

144 - 4-4 angle of decagonal prism

144.736 - 3-4 angle of rhombicuboctahedron, 3-4 angle of square cupola, 3-4 angle of elongated square pyramid, 3-4 angle of elongated square dipyramid
4-6 angle of truncated cuboctahedron
Relations: 54.7356 + 90

145.222 - 3-3 angle of hexagonal antiprism

148.283 - 4-5 angle of rhombicosidodecahedron, 4-5 angle of pentagonal cupola
4-10 angle of truncated icosidodecahedron

152.93 - 3-5 angle of snub dodecahedron

153.235 - 3-3 angle of snub cube

153.962 - 3-3 angle of octagonal antiprism

158.572 - join 3-3 angle of gyroelongated square pyramid, join 3-3 angle of gyroelongated square dipyramid
Relations: 54.7356 + 103.836

159.095 - 3-4 angle of rhombicosidodecahedron, 3-4 angle of pentagonal cupola
4-6 angle of truncated icosidodecahedron

159.187 - 3-3 angle of decagonal antiprism

160.523 - side 3-4 angle of elongated triangular cupola
Relations: +

160.529 - apex 3-4 angle of elongated triangular pyramid, 3-4 angle of elongated triangular dipyramid
Relations: 70.5288 + 90

164.172 - 3-3 angle of snub dodecahedron

Vertex data (and these are complete):

3-3-3: tetrahedron, elongated triangular pyramid, triangular dipyramid, elongated triangular dipyramid, augmented tridiminished icosahedron
3-3-4: square pyramid
3-3-5: pentagonal pyramid
3-4-4: triangular prism, elongated triangular pyramid, gyrobifastigium, augmented triangular prism
3-4-6: triangular cupola
3-4-8: square cupola
3-4-10: pentagonal cupola
3-5-5: metabidiminished icosahedron, tridiminished icosahedron, augmented tridiminished icosahedron, bilunabirotunda
3-5-10: pentagonal rotunda
3-6-6: truncated tetrahedron, augmented truncated tetrahedron
3-8-8: truncated cube, augmented truncated cube, biaugmented truncated cube
3-10-10: truncated dodecahedron, augmented truncated dodecahedron, parabiaugmented truncated dodecahedron, metabiaugmented truncated dodecahedron, triaugmented truncated dodecahedron
4-4-4: cube, elongated square pyramid
4-4-5: pentagonal prism, elongated pentagonal pyramid, augmented pentagonal prism, biaugmented pentagonal prism
4-4-6: hexagonal prism, elongated triangular cupola, augmented hexagonal prism, parabiaugmented hexagonal prism, metabiaugmented hexagonal prism
4-4-8: octagonal prism, elongated square cupola
4-4-10: decagonal prism, elongated pentagonal cupola, elongated pentagonal rotunda
4-5-10: diminished rhombicosidodecahedron, paragyrate diminished rhombicosidodecahedron, metagyrate diminished rhombicosidodecahedron, bigyrate diminished rhombicosidodecahedron, parabidiminished rhombicosidodecahedron, metabidiminished rhombicosidodecahedron, gyrate bidiminished rhombicosidodecahedron, tridiminished rhombicosidodecahedron
4-6-6: truncated octahedron
4-6-8: truncated cuboctahedron
4-6-10: truncated icosidodecahedron
5-5-5: dodecahedron, augmented dodecahedron, parabiaugmented dodecahedron, metabiaugmented dodecahedron, triaugmented dodecahedron
5-6-6: truncated icosahedron

3-3-3-3: octahedron, square pyramid, elongated square pyramid, gyroelongated square pyramid, triangular dipyramid, pentagonal dipyramid, elongated square dipyramid, gyroelongated square dipyramid, augmented triangular prism, biaugmented triangular prism, triaugmented triangular prism, augmented pentagonal prism, biaugmented pentagonal prism, augmented hexagonal prism, parabiaugmented hexagonal prism, metabiaugmented hexagonal prism, triaugmented hexagonal prism, snub disphenoid, augmented sphenocorona, sphenomegacorona
3-3-3-4: square antiprism, gyroelongated square pyramid, augmented triangular prism, biaugmented triangular prism, sphenocorona, augmented sphenocorona
3-3-3-5: pentagonal antiprism, gyroelongated pentagonal pyramid, metabidiminished icosahedron, tridiminished icosahedron, augmented tridiminished icosahedron, triangular hebesphenorotunda
3-3-3-6: hexagonal antiprism, gyroelongated triangular cupola
3-3-3-8: octagonal antiprism, gyroelongated square cupola
3-3-3-10: decagonal antiprism, gyroelongated pentagonal cupola, gyroelongated pentagonal rotunda
3-3-4-4: elongated triangular pyramid, elongated square pyramid, elongated pentagonal pyramid, elongated triangular dipyramid, elongated square dipyramid, elongated pentagonal dipyramid, triangular orthobicupola, square orthobicupola, pentagonal orthobicupola, sphenocorona, sphenomegacorona, hebesphenomegacorona, disphenocingulum
3-3-4-5: pentagonal gyrocupolarotunda, augmented pentagonal prism, biaugmented pentagonal prism
3-3-4-6: augmented hexagonal prism, parabiaugmented hexagonal prism, metabiaugmented hexagonal prism, triaugmented hexagonal prism, triangular hebesphenorotunda
3-3-5-5: pentagonal orthobirotunda, augmented dodecahedron, parabiaugmented dodecahedron, metabiaugmented dodecahedron, triaugmented dodecahedron, augmented tridiminished icosahedron
3-4-3-4: cuboctahedron, triangular cupola, elongated triangular cupola, gyroelongated triangular cupola, gyrobifastigium, triangular orthobicupola, square gyrobicupola, pentagonal gyrobicupola, elongated triangular orthobicupola, elongated triangular gyrobicupola, gyroelongated triangular bicupola, augmented truncated tetrahedron
3-4-3-5: pentagonal orthocupolarotunda, bilunabirotunda, triangular hebesphenorotunda
3-4-3-6: augmented truncated tetrahedron
3-4-3-8: augmented truncated cube, biaugmented truncated cube
3-4-3-10: augmented truncated dodecahedron, parabiaugmented truncated dodecahedron, metabiaugmented truncated dodecahedron, triaugmented truncated dodecahedron
3-4-4-4: rhombicuboctahedron, square cupola, elongated triangular cupola, elongated square cupola, elongated pentagonal cupola, gyroelongated square cupola, square orthobicupola, square gyrobicupola, elongated triangular orthobicupola, elongated triangular gyrobicupola, elongated square gyrobicupola, elongated pentagonal orthobicupola, elongated pentagonal gyrobicupola, elongated pentagonal orthocupolarotunda, elongated pentagonal gyrocupolarotunda, gyroelongated square bicopula, augmented truncated cube, biaugmented truncated cube
3-4-4-5: elongated pentagonal rotunda, elongated pentagonal orthocupolarotunda, elongated pentagonal gyrocupolarotunda, elongated pentagonal orthobirotunda, elongated pentagonal gyrobirotunda, gyrate rhombicosidodecahedron, parabigyrate rhombicosidodecahedron, metabigyrate rhombicosidodecahedron, trigyrate rhombicosidodecahedron, paragyrate diminished rhombicosidodecahedron, metagyrate diminished rhombicosidodecahedron, bigyrate diminished rhombicosidodecahedron, gyrate bidiminished rhombicosidodecahedron
3-4-5-4: rhombicosidodecahedron, pentagonal cupola, elongated pentagonal cupola, gyroelongated pentagonal cupola, pentagonal orthobicupola, pentagonal gyrobicupola, pentagonal orthocupolarotunda, pentagonal gyrocupolarotunda, elongated pentagonal orthobicupola, elongated pentagonal gyrobicupola, elongated pentagonal orthocupolarotunda, elongated pentagonal gyrocupolarotunda, gyroelongated pentagonal bicupola, gyroelongated pentagonal cupolarotunda, augmented truncated dodecahedron, parabiaugmented truncated dodecahedron, metabiaugmented truncated dodecahedron, triaugmented truncated dodecahedron, gyrate rhombicosidodecahedron, parabigyrate rhombicosidodecahedron, metabigyrate rhombicosidodecahedron, trigyrate rhombicosidodecahedron, diminished rhombicosidodecahedron, paragyrate diminished rhombicosidodecahedron, metagyrate diminished rhombicosidodecahedron, bigyrate diminished rhombicosidodecahedron, parabidiminished rhombicosidodecahedron, metabidiminished rhombicosidodecahedron, gyrate bidiminished rhombicosidodecahedron, tridiminished rhombicosidodecahedron
3-5-3-5: icosidodecahedron, pentagonal rotunda, elongated pentagonal rotunda, gyroelongated pentagonal rotunda, pentagonal orthocupolarotunda, pentagonal gyrocupolarotunda, pentagonal orthobirotunda, elongated pentagonal orthocupolarotunda, elongated pentagonal gyrocupolarotunda, elongated pentagonal orthobirotunda, elongated pentagonal gyrobirotunda, gyroelongated pentagonal cupolarotunda, gyroelongated pentagonal birotunda, bilunabirotunda, triangular hebesphenorotunda

3-3-3-3-3: icosahedron, pentagonal pyramid, elongated pentagonal pyramid, gyroelongated square pyramid, gyroelongated pentagonal pyramid, pentagonal dipyramid, elongated pentagonal dipyramid, gyroelongated square dipyramid, biaugmented triangular prism, triaugmented triangular prism, augmented dodecahedron, parabiaugmented dodecahedron, metabiaugmented dodecahedron, triaugmented dodecahedron, metabidiminished icosahedron, snub disphenoid, snub square antiprism, sphenocorona, augmented sphenocorona, sphenomegacorona, hebesphenomegacorona, disphenocingulum
3-3-3-3-4: snub cube, gyroelongated triangular cupola, gyroelongated square cupola, gyroelongated pentagonal cupola, gyroelongated triangular bicupola, gyroelongated square bicopula, gyroelongated pentagonal bicupola, gyroelongated pentagonal cupolarotunda, snub square antiprism, augmented sphenocorona, sphenomegacorona, hebesphenomegacorona, disphenocingulum
3-3-3-3-5: snub dodecahedron, gyroelongated pentagonal rotunda, gyroelongated pentagonal cupolarotunda, gyroelongated pentagonal birotunda
Marek14
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### Re: Johnsonian Polytopes

quickfur wrote:tacotopes

I don't like this at all.

Firstly, the names are a freakin' mouthful, no pun intended. Secondly, they're full of redundancy. "cubic ortho" and "antiprismatic gyro"? Each word in those terms implies the other.

You're not recognising the way I wanted to classify these shapes. I thought I made it clear that they are not formed by an operation adding one dimension to a 3D shape. They are formed by adding two dimensions to a 2D shape, just like the duoprisms. There is something called the hexagonal-pentagonal duoprism for example. Your system would probably call this a "hexagonal prism ortho-pentagonal prism tacotope", which is just silly.

The names for my series of 9 shapes need to be like this: (triangular|square|pentagonal) (ortho|gyro|magna)something. Even "square orthoringcupola", the logical name I'd devised and then rejected from it sounding bad, sounds better than your taco naming. The important parts of the name are the n-gon from which it is formed and the particular sub-operation being used to take that n-gon into 4D. In fact, I'll just go ahead and adjust my original name slightly, to "square orthobicupolic ring", which I have no problems with.

Similarly, for Mrrl's infinite family. The important thing is that from an n-gon a ring of two antiprisms and a prism are assembled to form the resulting duoprism-like shape. In the same vein as my ring polychora, these should be named "n-gonal biantiprismatic ring". Notice I didn't even include the prism in the name, it is implied by the fact that it is a ring.

It's also worth pointing out that you shouldn't be naming these tacotopes based on how the projection happens to look a bit like a sandwich. The shapes look nothing like sandwiches in 4D.

Keiji

Posts: 1969
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

### Re: Johnsonian Polytopes

Keiji wrote:[...]
It's also worth pointing out that you shouldn't be naming these tacotopes based on how the projection happens to look a bit like a sandwich. The shapes look nothing like sandwiches in 4D.

No, I didn't base the name on the projection, but on the 4D shape itself. It consists of two facets joined at the wide bases, with a facet of a different kind connecting their two narrow tops, forming a triangular configuration with two equal members and one unequal member. This is the basic form of the shape; the rest of the cells are added by the convex hull, and the added cells basically lie around the one facet of a different kind and between the two facets of the same kind.

And yes, the name was a bit silly but I was trying to capture this A-A-B triangular ring structure that forms the basis of these shapes. I wasn't trying to add a dimension to a 3D shape; I was trying to describe the 4D construction process using an analogy with how a widely-known object in 3D is similarly constructed: take a "shell" consisting of two A's joined at their wide bases, and fold them up in 4D and stick a B at the narrow end to make a 3-membered ring, then fill in the remaining gaps with other miscellaneous pieces.
quickfur
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Location: The Great White North

### Re: Johnsonian Polytopes

Alright, I was bored this morning so I decided to calculate the coordinates for the triangular cupola version of Keiji's CRF polychora. So here's the ortho variant:

Enjoy!

EDIT: and the coordinates are:

Hexagon:
(±sqrt(3), ±1, 0, 0)
(0, ±2, 0, 0)
Triangles:
(-1/sqrt(3), ±1, ±1, sqrt(5/3))
(2/sqrt(3), 0, ±1, sqrt(5/3))
Last edited by quickfur on Tue Nov 22, 2011 5:36 pm, edited 1 time in total.
quickfur
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Location: The Great White North

### Re: Johnsonian Polytopes

You should really get these images on the wiki sometime so they can get decent pages.

Keiji

Posts: 1969
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

### Re: Johnsonian Polytopes

Keiji wrote:You should really get these images on the wiki sometime so they can get decent pages.

Alright, I was going to work on the coordinates for the gyro 3-prism cupola tritope (still trying to come up with a satisfactory name for these things), but I think i should start uploading all these renders to the wiki instead, 'cos i'm starting to accumulate too many temporary renders on my server. I think from now on I should just upload them to the wiki; it's a bad idea to have forum posts depending on external images in the first place.
quickfur
Pentonian

Posts: 2841
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

quickfur wrote:[...](still trying to come up with a satisfactory name for these things)[...]

Actually, I like your latest terminology on the wiki (bicupolic rings and biantiprismic rings). I'll stick to that from now.
quickfur
Pentonian

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Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

OK, here's the triangular gyrobicupolic ring:

Again I turned off visibility clipping because this polytope is relatively simple. The cells are easy to pick out: 6 square pyramids, 1 octahedron, and two triangular cupola.

The coordinates I'm using are:

Hexagon:
(±sqrt(3), ±1, 0, 0)
(0, ±2, 0, 0)
Triangle 1:
(-1/sqrt(3), ±1, sqrt(2/3), sqrt(2))
(2/sqrt(3), 0, sqrt(2/3), sqrt(2))
Triangle 2:
(1/sqrt(3), ±1, -sqrt(2/3), sqrt(2))
(-2/sqrt(3), 0, -sqrt(2/3), sqrt(2))
quickfur
Pentonian

Posts: 2841
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

I see you just uploaded this:

Could you render it with a cubical envelope, i.e. the rotated square face in the middle? I think that would nicer, since the projection would be more symmetrical and easier to interpret.

Keiji

Posts: 1969
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

### Re: Johnsonian Polytopes

I was going to do multiple renders of this one, because none of the usual views are very easy to interpret. At the very least, I would highlight some of the cells in order to show the structure better.

But how do I rename that file? "test.png" is a totally stupid name.
quickfur
Pentonian

Posts: 2841
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Location: The Great White North

### Re: Johnsonian Polytopes

In Kawachan, files don't have names, they only have hashes. There's no way I or anyone else would have known it was called "test.png" if you hadn't just said so.

Keiji

Posts: 1969
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

### Re: Johnsonian Polytopes

That's what I thought, I guess I got thrown off by the post-upload page that says "successfully uploaded test.png", so i thought it was actually going to use that lousy name. Sighh....
quickfur
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Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

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