Johnsonian Polytopes

[Keiji]

Alright folks, the tarballs are ready
...
P.S. These files are released under GPL3. Just FYI.

Hooray

I'll take a look a little later, when I'm not so busy with other things.

BTW Keiji, I can't seem to upload files anymore; I'm getting a page that says "Error calling file_dyn".

I retired a part of my server earlier, forgetting that the file uploading scripts happened to depend on it. I've just changed the important path to where it probably should have been all along, so it's fixed now. As always, there's just one little thing you forget when doing something like this

Edit: and apparently my avatar was hosted there as well! Good thing there are backups.

[quickfur]

Alright folks, the tarballs are ready:

Just realized I didn't include the latest CRF polytope definitions with the data tarball. I've uploaded them as a separate tarball (since they're an ongoing project; so you don't have to keep redownloading the data tarball):

http://eusebeia.dyndns.org/polyview/crf-data.tar.gz

The tarball includes the source data in Polyhedra format (exactly what I passed to makepoly to make the .def files).

[Marek14]

[...] As for color algorithm, I already offered one: make a list of color definitions, and assign the same color to cells with the same number of faces. More complex would be to distinguish between cells with different kinds of faces (like checking numbers for 5-faced cells and use one color if it has 2 triangles and 3 squares and another if it has 4 triangles and 1 square, but this should be enough.

Also, a good idea would probably be to separate the face and cell date in the file by comment line. [...]

OK, I've thought of a clever way to make color assignments of different cell types unique. It just so happens that in 4D, cells have 3 types of surtopes: vertices, edges, and ridges. So each cell type will have an almost unique combination of number of vertices, edges, and ridges. So this can be mapped to RGB values.

But direct mapping doesn't produce good results (stepping by 1 in RGB value is invisible to human eye). So take the sine or cosine of the number and then scale it to fit the range 0-255. Because integers are incommensurable with pi, the sine of an integer will generate a non-repeating but periodic sequence (period is irrational so numbers will not repeat) which is always bounded between 0-255. Not scaling the integer also means we cycle from 0 to 255 with integer argument 0-4, so there will be large constrast between similar but non-identical cells. So this should produce good results. I hope.

Try it:

This works. Actually, once I open it in Stella, I can order recoloring, changing the colors to more bright primaries, and then save it as *.stel file. (Stella can distinguish between two cells that have the same shape, but different functions within the polychoron, and assign them different colors, which is useful.)

As for truncated tetrahedron||truncated octahedron, you ARE aware that segmentochora were completely enumerated, right? This one is 4.76:

4.76 truncated tetrahedron ||
truncated octahedron
height: sqrt(5/8) = 0.790569
circumradius: sqrt(13/5) = 1.612452
other names: truncated-tetrahedral monostratic
cup of runcinated pentachoron, truncatedtetrahedral
monostratic cup of prismatorhombated
pentachoron
cells: 6 trigonal prisms + 4 trigonal
cupolae + 4 hexagonal prisms + 1 truncated
tetrahedron + 1 truncated octahedron

[Marek14]

Messing with Stella, I got this:

Code: Select all

OFF
# C100_010_001_bicupola
# Generated by 'Stella4D', Version 4.4
# Author: Robert Webb
# Web site: http://www.software3d.com/Stella.php
# Licensed to: Marek E`trnáct
# Date: 23:34:22, 24 November 2011
#
# Copyright (C) Robert Webb, 2001-2008.
# This copyright notice and the above
# information must be kept intact.
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3 126 124 125    0 255 0
3 127 128 129    0 255 0
3 127 128 130    0 255 0
3 127 129 131    0 255 0
3 127 130 131    0 255 0
3 128 129 132    0 255 0
3 128 132 130    0 255 0
3 129 132 131    0 255 0
3 132 130 131    0 255 0
3 133 134 135    0 255 0
3 133 134 136    0 255 0
3 133 135 137    0 255 0
3 133 136 137    0 255 0
3 134 135 138    0 255 0
3 134 138 136    0 255 0
3 135 138 137    0 255 0
3 138 136 137    0 255 0
4 139 140 141 142    255 255 0
3 139 140 143    255 255 0
3 139 142 143    255 255 0
3 141 140 143    255 255 0
3 141 142 143    255 255 0
3 144 145 146    0 255 0
3 144 145 147    0 255 0
3 144 146 148    0 255 0
3 144 147 148    0 255 0
3 145 146 149    0 255 0
3 145 149 147    0 255 0
3 146 149 148    0 255 0
3 149 147 148    0 255 0
3 150 151 152    0 255 0
3 150 151 153    0 255 0
3 150 152 154    0 255 0
3 150 153 154    0 255 0
3 151 152 155    0 255 0
3 151 155 153    0 255 0
3 152 155 154    0 255 0
3 155 153 154    0 255 0
4 156 157 158 159    255 255 0
3 156 157 160    255 255 0
3 156 159 160    255 255 0
3 158 157 160    255 255 0
3 158 159 160    255 255 0
4 161 162 163 164    255 255 0
3 161 162 165    255 255 0
3 161 164 165    255 255 0
3 163 162 165    255 255 0
3 163 164 165    255 255 0


It's one of the 4D views of the object converted to 3D scene. Unfortunately, it seems I can't save 4D scene as OFF...

[quickfur]

[...] As for truncated tetrahedron||truncated octahedron, you ARE aware that segmentochora were completely enumerated, right? This one is 4.76:[...]

Yes I know. I glanced at the paper you referenced (didn't read it in detail though). Although personally, I am trying to gain a better understanding of how these shapes are generated, so that hopefully i can get some ideas about constructing shapes that are not on Klietzing's list.

I think I have quite a good handle on segmentochora with uniform top/bottom facets now. It's still fun to explore the ones already known, though, some of them have interesting antiprism cells (like dodecahedron||icosidodecahedron, with pentagonal antiprisms).

However, it hasn't been easy to imagine shapes outside this set. Keiji found the cupolic rings with 3-membered cycles, and most of the segmentochora can be regarded as having 4-membered cycles of cells, but what about more complex arrangements? Such as 5- or 6- membered cycles built from Johnson solids. Those are hard to discover, IMHO harder than discovering the Johnson solids themselves, because there are only so many ways to put regular polygons together: any edge can fit with any other edge, so edges have only one kind of "function", but in 4D we have all the irregular shapes inherited from 3D johnson solids which have ridges that have many different functions (ridges have all kinds of shapes and most of the time they don't fit together).

[quickfur]

Messing with Stella, I got this:
[...]
It's one of the 4D views of the object converted to 3D scene. Unfortunately, it seems I can't save 4D scene as OFF...

Do you have an image of it? I don't have stella.

[Marek14]

Well, that's why I exported it in OFF format... I thought you'll be able to view it

Stella is capable of 3D export in DXF, POV-Ray, VRML and OBJ. But, here are two views as normal pictures

C100_010_001_bicupola.gif (21.24 KiB) Viewed 374483 times

C100_010_001_bicupola_2.gif (20.27 KiB) Viewed 374483 times

[quickfur]

Well, that's why I exported it in OFF format... I thought you'll be able to view it

Well, in theory I could write a convertor to generate a povray scene from it, I suppose... But I'm thinking more about how to integrate the various polychoron generators that I currently have as separate programs, so that a single program can generate all those kinds of shapes, as well as mixing some of the functionality, etc.. I'm also thinking of how to automatically generate models of the segmentochora (as well as tell you which combinations are invalid, etc.).

Stella is capable of 3D export in DXF, POV-Ray, VRML and OBJ. But, here are two views as normal pictures

Eeek... these Schlegel-diagram type projections give me a headache. I feel more comfortable with perspective projection at a distance. But anyway, I've already posted renders of this polychoron earlier in this topic.

[Mrrl]

There is one more kind of connections between rectified dodecahedron and truncated dodecahedron: instead of using of 12 pentagonal cupolae and 20 tetrahedra, you can take 12 pentagonal rotundae and 40 tetrahedra! It will be a cup that you cut off when diminish rectified 120-cell.
So diminished rectified 120-cells (but there are only few of them, the closest pair of cells that you can cut off is cells connected to the opposite sides of one cell, theoretical limit of cells that can be cut off is [120/7]=17, but we'll be lucky if find 12 such cells) contains icosadodecahedra, pentagonal rotundae and tetrahedra as its cells.
Another interesting fact is with cantellated 24-cells. There are two levels of diminishing centered in cuboctahedral cell: you may cut off either truncated cube||cuboctahedron cupola, or truncated cuboctahedron||truncated cube||cuboctahedron sequence. It gives us nothing new in terms of elementary CRFs, but looks interesting.

[Marek14]

Well, that's why I exported it in OFF format... I thought you'll be able to view it

Well, in theory I could write a convertor to generate a povray scene from it, I suppose... But I'm thinking more about how to integrate the various polychoron generators that I currently have as separate programs, so that a single program can generate all those kinds of shapes, as well as mixing some of the functionality, etc.. I'm also thinking of how to automatically generate models of the segmentochora (as well as tell you which combinations are invalid, etc.).

Stella is capable of 3D export in DXF, POV-Ray, VRML and OBJ. But, here are two views as normal pictures

Eeek... these Schlegel-diagram type projections give me a headache. I feel more comfortable with perspective projection at a distance. But anyway, I've already posted renders of this polychoron earlier in this topic.

There are already VRML models of all segmentochora here: http://www.orchidpalms.com/polyhedra/se ... /table.htm
Maybe they could be converted? I don't know much about this format.

[quickfur]

There is one more kind of connections between rectified dodecahedron and truncated dodecahedron: instead of using of 12 pentagonal cupolae and 20 tetrahedra, you can take 12 pentagonal rotundae and 40 tetrahedra! It will be a cup that you cut off when diminish rectified 120-cell.

Hmm, you're right!! I had totally forgotten that rectified dodecahedron can be cut in half while remaining CRF. Do you think there's any chance that we can cut the rectified 120-cell at an even deeper level? Is there any possibility of bisecting the rectified 120-cell to get a CRF? It's not possible with 120-cell itself because the dodecahedron can't be bisected without losing regular faces, but maybe it's possible with the rectified 120-cell?

[...] Another interesting fact is with cantellated 24-cells. There are two levels of diminishing centered in cuboctahedral cell: you may cut off either truncated cube||cuboctahedron cupola, or truncated cuboctahedron||truncated cube||cuboctahedron sequence. It gives us nothing new in terms of elementary CRFs, but looks interesting.

That might be worth rendering if we can compute coordinates.

[quickfur]

[...] There are already VRML models of all segmentochora here: http://www.orchidpalms.com/polyhedra/se ... /table.htm
Maybe they could be converted? I don't know much about this format.

I doubt it. VRML is (was?) a language they wanted to become the standard 3D modelling format on the web, but for various reasons it didn't really take off. But it's only a 3D modelling format; I don't think you can extract 4D data from it. Or maybe you can use brute-force methods by identifying top/bottom cells and using floating-point arithmetic to make edge lengths equal. But most likely it won't give us nice algebraic coordinates.

[quickfur]

There is one more kind of connections between rectified dodecahedron and truncated dodecahedron: instead of using of 12 pentagonal cupolae and 20 tetrahedra, you can take 12 pentagonal rotundae and 40 tetrahedra! It will be a cup that you cut off when diminish rectified 120-cell.

Hmm, you're right!! I had totally forgotten that rectified dodecahedron can be cut in half while remaining CRF. Do you think there's any chance that we can cut the rectified 120-cell at an even deeper level? Is there any possibility of bisecting the rectified 120-cell to get a CRF? It's not possible with 120-cell itself because the dodecahedron can't be bisected without losing regular faces, but maybe it's possible with the rectified 120-cell?
[...]

P.S. I think your cut is actually something outside the segmentotope set, because, if i'm not mistaken, there are two sets of tetrahedra at two different heights in the resulting cup-shaped CRF! So this is a new kind of CRF polychoron!

But as for a deeper cut of the rectified 120-cell, I'm not 100% sure yet but I think it's probably not possible, because once you pass the first layer of 12 rectified dodecahedra, the next layer of rectified dodecahedra have two different orientations, so most likely they won't cut "cleanly" into Johnson polyhedra.

[Mrrl]

P.S. I think your cut is actually something outside the segmentotope set, because, if i'm not mistaken, there are two sets of tetrahedra at two different heights in the resulting cup-shaped CRF! So this is a new kind of CRF polychoron!

Yes, it's dodecahedral rotunda I purposefully looked for this kind of polytopes

But as for a deeper cut of the rectified 120-cell, I'm not 100% sure yet but I think it's probably not possible, because once you pass the first layer of 12 rectified dodecahedra, the next layer of rectified dodecahedra have two different orientations, so most likely they won't cut "cleanly" into Johnson polyhedra.

It's not possible for more simple reason: you can get only 3-,5- and 10-gons in the section that gives us CRF, and there is no larger CRF polyhedron than truncated dodecahedron...

I've took a look at the Klitzing list. Even in the begining there is something that I can't recognize:

Code: Select all

4.8 trigon || square pyramid
height: sqrt(5/8) = 0.790569
shear: 1/sqrt(24) = 0.204124
4.8.1 square || tetrahedron
height: sqrt(5/8) = 0.790569
shear: 0
4.8.2 line || orthogonal trigonal prism
height: sqrt(5/12) = 0.645497
shear: 1/sqrt(12) = 0.288675
circumradius: sqrt(3/5) = 0.774597
other names: trigonal square-pyramidal wedge
comments: kind of bidiminished rectifiedpentachoron
(tetrahedron as "tetrahedron - 2 edges"
and square as "octahedron - 2 square pyramids")
cells: 1 tetrahedron + 4 square pyramids
+ 2 trigonal prisms

And close to the end there is a set of biantiprismatic rings (4.174), he calls them "general n-gonal n-antiprismatic
wedge"

[quickfur]

P.S. I think your cut is actually something outside the segmentotope set, because, if i'm not mistaken, there are two sets of tetrahedra at two different heights in the resulting cup-shaped CRF! So this is a new kind of CRF polychoron!

Yes, it's dodecahedral rotunda I purposefully looked at this kind of polytopes

Now we need coordinates for it.

Do you think it's possible to runcinate this to get another rotunda? Something with dodecahedron at the top with 12 pentagonal prisms connected to the pentagonal rotunda, with cuboctahedra instead of tetrahedra and probably more prisms to fill in the remaining gaps. Does it exist?

[...] It's not possible for more simple reason: you can get only 3-,5- and 10-gons in the section that gives us CRF, and there is no larger CRF polyhedron than truncated dodecahedron...

OK.

I've took a look at the Klitzing list. Even in the begining there is something that I can't recognize:

Code: Select all

4.8 trigon || square pyramid
height: sqrt(5/8) = 0.790569
shear: 1/sqrt(24) = 0.204124
4.8.1 square || tetrahedron
height: sqrt(5/8) = 0.790569
shear: 0
4.8.2 line || orthogonal trigonal prism
height: sqrt(5/12) = 0.645497
shear: 1/sqrt(12) = 0.288675
circumradius: sqrt(3/5) = 0.774597
other names: trigonal square-pyramidal wedge
comments: kind of bidiminished rectifiedpentachoron
(tetrahedron as "tetrahedron - 2 edges"
and square as "octahedron - 2 square pyramids")
cells: 1 tetrahedron + 4 square pyramids
+ 2 trigonal prisms

Hmm that's interesting. If I understand his notation correctly, he has a square pyramid at the bottom and a triangle at the top? Can that have equal length edges?

And close to the end there is a set of biantiprismatic rings (4.174), he calls them "general n-gonal n-antiprismatic
wedge"

Hmm, this looks like the antiprismatic rings you found. I suspect that Keiji's bicupolic rings may also be included in the "wedge" category. Is there a way to confirm this?

[Mrrl]

And close to the end there is a set of biantiprismatic rings (4.174), he calls them "general n-gonal n-antiprismatic
wedge"

Hmm, this looks like the antiprismatic rings you found. I suspect that Keiji's bicupolic rings may also be included in the "wedge" category. Is there a way to confirm this?

These ones?

Code: Select all

4.133 pentagon || gyrated
pentagonal cupola
height: 1/2
shear: sqrt((5+2*sqrt(5))/5) = 1.376382
4.133.1 decagon || pentagonal
antiprism
height: (sqrt(5)-1)/4 = 0.309017
shear: 0
circumradius: sqrt(5+2*sqrt(5)) = 3.077684
other names: -
comments: kind of diminished
icosidodecahedral-cupola (pentagon as
"icosidodecahedron - icosidodecahedron" and
pentagonal cupola as "rhombicosidodecahedron -
diminished rhombicosidodecahedron") -resp.- kind
of bidiminished icosahedral-cupola (decagon as
"icosidodecahedra - 2 rotunda" and pentagonal
antiprism as "icosahedron - 2 pentagonal
pyramids")
cells: 10 square pyramids + 1
pentagonal antiprism + 2 pentagonal cupolae

Code: Select all

4.154 pentagon || pentagonal cupola
height: (sqrt(5)-1)/4 = 0.309017
shear: sqrt((25+11*sqrt(5))/40) =
1.113516
4.154.1 decagon || pentagonal
prism
height: sqrt((5-2*sqrt(5))/20) = 0.162460
shear: 0
circumradius: 3+sqrt(5) = 5.236068
other names: pentagonal pentagonal-cupolaic
wedge
comments: kind of diminished
dodecahedronal-monostratic-cup-of-runcinatedhecatonicosachoron
(pentagon as "dodecahedron -
dodecahedron" and pentagonal cupola as
"rhombicosidodecahedron - diminished
rhombicosidodecahedron (J76)")
cells: 5 tetrahedra + 5 trigonal prisms +
1 pentagonal prism + 2 pentagonal cupolae

Code: Select all

4.165 decagon || pentagonal cupola
height: (sqrt(5)-1)/4 = 0.309017
shear: sqrt((25+11*sqrt(5))/8) =
2.489893
4.165.1 pentagon || decagonal
prism
height: sqrt((5-2*sqrt(5))/20) = 0.162460
shear: 0
circumradius: sqrt(23+10*sqrt(5)) = 6.735034
other names: decagonal pentagonal-cupolaic
wedge
comments: kind of diminished
rhombicosidodecahedral-monostratic-cup-ofcantellated-
hecatonicosachoron (pentagonal cupola
as "rhombicosidodecahedron - diminished
rhombicosidodecahedron (J76)" and decagon as
"truncated dodecahedron - truncated
dodecahedron")
cells: 5 square pyramids + 5 trigonal
prisms + 1 decagonal prism + 2 pentagonal cupolae

[quickfur]

[...]

Code: Select all

4.133 pentagon || gyrated
pentagonal cupola
height: 1/2
shear: sqrt((5+2*sqrt(5))/5) = 1.376382
4.133.1 decagon || pentagonal
antiprism
height: (sqrt(5)-1)/4 = 0.309017
shear: 0
circumradius: sqrt(5+2*sqrt(5)) = 3.077684
other names: -
comments: kind of diminished
icosidodecahedral-cupola (pentagon as
"icosidodecahedron - icosidodecahedron" and
pentagonal cupola as "rhombicosidodecahedron -
diminished rhombicosidodecahedron") -resp.- kind
of bidiminished icosahedral-cupola (decagon as
"icosidodecahedra - 2 rotunda" and pentagonal
antiprism as "icosahedron - 2 pentagonal
pyramids")
cells: 10 square pyramids + 1
pentagonal antiprism + 2 pentagonal cupolae

Yeah, this looks like what Keiji calls the pentagonal gyrobirotundic ring.

Code: Select all

4.154 pentagon || pentagonal cupola
height: (sqrt(5)-1)/4 = 0.309017
shear: sqrt((25+11*sqrt(5))/40) =
1.113516
4.154.1 decagon || pentagonal
prism
height: sqrt((5-2*sqrt(5))/20) = 0.162460
shear: 0
circumradius: 3+sqrt(5) = 5.236068
other names: pentagonal pentagonal-cupolaic
wedge
comments: kind of diminished
dodecahedronal-monostratic-cup-of-runcinatedhecatonicosachoron
(pentagon as "dodecahedron -
dodecahedron" and pentagonal cupola as
"rhombicosidodecahedron - diminished
rhombicosidodecahedron (J76)")
cells: 5 tetrahedra + 5 trigonal prisms +
1 pentagonal prism + 2 pentagonal cupolae

And this one is the pentagonal orthobicupolic ring. And actually, I like Klitzing's term "wedge" better, 'cos that's what these things are, 4D wedges of various shapes.

Code: Select all

4.165 decagon || pentagonal cupola
height: (sqrt(5)-1)/4 = 0.309017
shear: sqrt((25+11*sqrt(5))/8) =
2.489893
4.165.1 pentagon || decagonal
prism
height: sqrt((5-2*sqrt(5))/20) = 0.162460
shear: 0
circumradius: sqrt(23+10*sqrt(5)) = 6.735034
other names: decagonal pentagonal-cupolaic
wedge
comments: kind of diminished
rhombicosidodecahedral-monostratic-cup-ofcantellated-
hecatonicosachoron (pentagonal cupola
as "rhombicosidodecahedron - diminished
rhombicosidodecahedron (J76)" and decagon as
"truncated dodecahedron - truncated
dodecahedron")
cells: 5 square pyramids + 5 trigonal
prisms + 1 decagonal prism + 2 pentagonal cupolae

And this is the pentagonal magnabirotundic ring/wedge.

Heh heh... so it looks like the only "new" discovery we made so far is your dodecahedral rotunda.

[quickfur]

I'm thinking about a possible CRF, but not sure if it's valid: start with a tetrahedron, then attach 4 elongated triangular cupola, and insert triangular prisms at the elongated part of the cupolae to close up the shape, then the bottom cell is a truncated octahedron. There may be another kind of prism between the cupola part of the elongated cupolae, i think either cubes?

Can someone check if this is constructible? I'm not sure whether the triangular prisms inserted at the elongated part of the cupolae can be made uniform.

[Mrrl]

So... what classes of CRF do we have now?
Uniform polychora (including 2 infinite families)
Segmentotopes (including +1 infinine family)
Dodecahedric rotunda
Chains of segmentotopes+rotunda (there is finite number of them and we can enumerate them)
Diminished, double-diminished and rotunda-diminished uniform polychora (finite set)
CRFs listed above recursively augmented by segmentotopes or rotunda.
First question: is there any convex combination of the couples of bodies listed above?
Second: There is a large gap between augmented chains of segmetotopes and diminished forms from 24- and 120/600-cells families. What is in this gap?

[quickfur]

I'm thinking about a possible CRF, but not sure if it's valid: start with a tetrahedron, then attach 4 elongated triangular cupola, and insert triangular prisms at the elongated part of the cupolae to close up the shape, then the bottom cell is a truncated octahedron. [...]

Actually, I just realized that a truncated octahedron is tiled by a tetrahedron and 4 triangular cupola (plus a few other pieces), so the cupola (elongated or otherwise) lie on a flat hyperplane.

[quickfur]

[...] Dodecahedric rotunda

I was just thinking about this... are you sure the top cell is a dodecahedron? I thought it should be a rectified dodecahedron.

Also, I'm more confident now that it is possible to runcinate this rotunda, you will get the same count of rectified dodecahedra, but with pentagonal prisms between them, and the tetrahedra around the top cell become cuboctahedra, and the tetrahedra around the bottom cell become triangular cupola, and the bottom cell is expanded into great rhombicosidodecahedron. It is essentially the diminishing of a cantellated 600-cell (CD diagram is o5x3o3x), some of the cuboctahedra are cut into triangular cupolae.

Chains of segmentotopes+rotunda (there is finite number of them and we can enumerate them)
Diminished, double-diminished and rotunda-diminished uniform polychora (finite set)
CRFs listed above recursively augmented by segmentotopes or rotunda.
First question: is there any convex combination of the couples of bodies listed above?

What do you mean by couples of bodies? I thought we already included recursive augmentations of other CRFs by segmentotopes and rotunda?

Second: There is a large gap between augmented chains of segmetotopes and diminished 24- and 120/600-cells. What is inside?

You mean what is in between?

It's hard to tell... I think there must be some fundamental CRFs that cannot be generated by cutting uniform polytopes or augmenting by segmentotopes. I noticed in 3D johnson solids there are some fundamental shapes that have parts that look like subset of icosahedron's faces, but with modifications or substitution by other faces. I wonder if we can start with icosahedral configuration of tetrahedra (as the simplest subset of the 600-cell that captures its configuration of tetrahedral cells), and see if we can make a closed shape by replacing parts of it with other polyhedra?

[Marek14]

Code: Select all

4.8 trigon || square pyramid
height: sqrt(5/8) = 0.790569
shear: 1/sqrt(24) = 0.204124
4.8.1 square || tetrahedron
height: sqrt(5/8) = 0.790569
shear: 0
4.8.2 line || orthogonal trigonal prism
height: sqrt(5/12) = 0.645497
shear: 1/sqrt(12) = 0.288675
circumradius: sqrt(3/5) = 0.774597
other names: trigonal square-pyramidal wedge
comments: kind of bidiminished rectifiedpentachoron
(tetrahedron as "tetrahedron - 2 edges"
and square as "octahedron - 2 square pyramids")
cells: 1 tetrahedron + 4 square pyramids
+ 2 trigonal prisms

I made an OFF file out of this:

Code: Select all

4OFF
# NumVertices, NumFaces, NumEdges, NumCells
8 17 16 7

# Vertices
0    0.57735026918962576 -0.5 0
-0.5 -0.28867513459481288 -0.5 0
0.5 -0.28867513459481288 -0.5 0
0    0.57735026918962576  0.5 0
-0.5 -0.28867513459481288  0.5 0
0.5 -0.28867513459481288  0.5 0
-0.5  0.28867513459481288  0   0.64549722436790281
0.5  0.28867513459481288  0   0.64549722436790281

# Faces

3 0 1 2
3 3 4 5
4 0 1 4 3
4 0 2 5 3
4 1 2 5 4

3 0 1 6
3 0 2 7
4 1 2 7 6
3 3 4 6
3 3 5 7

4 4 5 7 6
3 0 3 6
3 0 3 7
3 1 4 6
3 2 5 7

3 0 6 7
3 3 6 7

# Cells

5  0  1  2  3  4    0   0 255
5  0  5  6  7 15    0 255   0
5  1  8  9 10 16    0 255   0
5  2  5  8 11 13    0 255   0
5  3  6  9 12 14    0 255   0
5  4  7 10 13 14    0   0 255
4 11 12 15 16     255   0   0


Here are the three segmentochoric projections given by the Klitzing's constructions:

4.8.gif (9.16 KiB) Viewed 374466 times

4.8.1.gif (10.65 KiB) Viewed 374466 times

4.8.2.gif (10.96 KiB) Viewed 374466 times

[Keiji]

Well that went well:

Code: Select all

hayate@nanako+0:proglib$ scons
scons: Reading SConscript files ...

scons: *** Two environments with different actions were specified for the same target: cfgfile.o
File "/home/hayate/archive/bin/polyview/proglib/c++/SConscript", line 30, in <module>


[quickfur]

Well that went well:

Code: Select all

hayate@nanako+0:proglib$ scons
scons: Reading SConscript files ...

scons: *** Two environments with different actions were specified for the same target: cfgfile.o
File "/home/hayate/archive/bin/polyview/proglib/c++/SConscript", line 30, in <module>


Huh, weird. I just tested the tarball myself, and it seems fine. Which version of SCons are you using? (What's the output of scons --version?)

[Mrrl]

[...] Dodecahedric rotunda

I was just thinking about this... are you sure the top cell is a dodecahedron? I thought it should be a rectified dodecahedron.

Yes, there is rectified dodecahedron.

Also, I'm more confident now that it is possible to runcinate this rotunda, you will get the same count of rectified dodecahedra, but with pentagonal prisms between them, and the tetrahedra around the top cell become cuboctahedra, and the tetrahedra around the bottom cell become triangular cupola, and the bottom cell is expanded into great rhombicosidodecahedron. It is essentially the diminishing of a cantellated 600-cell (CD diagram is o5x3o3x), some of the cuboctahedra are cut into triangular cupolae.

Yes, I also think that it exists. More, you can cut it in one more level and get great rhombicosidodecahedron|||truncated icosahedron rotunda It has 12 pentagonal rotunda, 30 pentagonal prisms and 40 triangular cupola as side cells

Chains of segmentotopes+rotunda (there is finite number of them and we can enumerate them)
Diminished, double-diminished and rotunda-diminished uniform polychora (finite set)
CRFs listed above recursively augmented by segmentotopes or rotunda.
First question: is there any convex combination of the couples of bodies listed above?

What do you mean by couples of bodies? I thought we already included recursive augmentations of other CRFs by segmentotopes and rotunda?

Like augmentations of something by side cell of segmentotope (that can't be used as base because other side is not parallel to this cell), or, better, combining two segmentotopes by cells (at least one of which is not a base) that have some bichoral angles >90 (like we tried to do with two biantiprasmatic rings). All augmentations can be done after that.

[Keiji]

Well that went well:

Huh, weird. I just tested the tarball myself, and it seems fine. Which version of SCons are you using? (What's the output of scons --version?)

Code: Select all

hayate@nanako+0:~$ scons --version
SCons by Steven Knight et al.:
        engine: v0.97.0d20071203.r2509, 2007/12/03 20:20:38, by broonie on mercator
Copyright (c) 2001, 2002, 2003, 2004, 2005, 2006, 2007 The SCons Foundation


Old sure, but why does everyone have to insist on using new solutions to the same old problems like compiling a C program? make works perfectly fine, there is no reason to use something like scons.
This particular server is using Ubuntu 8.04 LTS, so it's still supported for a year and a half.

I'll try it on my TS later, which runs 11.04, but the point still remains.

[quickfur]

[...]

Code: Select all

hayate@nanako+0:~$ scons --version
SCons by Steven Knight et al.:
        engine: v0.97.0d20071203.r2509, 2007/12/03 20:20:38, by broonie on mercator
Copyright (c) 2001, 2002, 2003, 2004, 2005, 2006, 2007 The SCons Foundation


Old sure, but why does everyone have to insist on using new solutions to the same old problems like compiling a C program?

Actually it's not that I insist on using new solutions... I don't have the older version of scons so I'm unaware when something no longer works on the old version. I'll see if I can fix this particular incompatibility, it shouldn't be too hard.

make works perfectly fine, there is no reason to use something like scons.

If you've ever done anything non-trivial with make, you'll hate it just as much as I do. Full of loopholes and inconsistencies and odd behaviours for anything more than textbook examples, and very unreliable.

Ever seen a project that has 'make clean', 'make realclean', 'make distclean', and who knows what else? Why is it even necessary to clean the workspace in the first place? Shouldn't the build system know how to rebuild a project correctly without you having to babysit it every time? Ever added a comment to a header file and have the entire project rebuild in 1.5 hours? Ever make a change and run 'make', and spend 6 hours debugging your program only to discover at the end that the code was correct, only make failed to rebuild a particular target? Worse yet, ever work over NFS and have make always rebuild the entire 2.5 million line codebase just because the NFS server's clock is slightly off from your workstation? What's the point of a build system if you have to rebuild the whole damn thing every single time anyway? Why not just use a shell script of gcc commands? Then you have people who use scripts and makefile hacks to coax it to do what they want, and the amount of makefile code exceeds the amount of source code? (I'm not kidding, my work project used to have something like this before they cleaned it up.) And most of the time, these hacks are just houses of cards with global variables and whatnot that break subtly every time you try to do anything with it. Most of the time you don't notice the bugs until it comes back to bite you much later. And don't even get me started on parallel builds. Make is at least 3 decades old and decrepit and is overdue for a long-deserved death. Computing has advanced way past its dinosaur age, there's no reason to keep clinging to the past.

Sorry, didn't mean to rant there. But you did touch a nerve.

This particular server is using Ubuntu 8.04 LTS, so it's still supported for a year and a half.

I'll try it on my TS later, which runs 11.04, but the point still remains.

[quickfur]

[...] I'll see if I can fix this particular incompatibility, it shouldn't be too hard.

OK, try this: http://eusebeia.dyndns.org/polyview/polyview-3.0pre-3561.tar.gz

Actually, if you see any more problems, i want to know about it, 'cos i do try to keep my code as cross-platform as possible (well, at least across POSIX systems ) and any cross platform problems are bugs that should be fixed.

[Keiji]

I extracted that over the same directory and I get the same error.

If I run scons in the polyview directory (instead of proglib), I get this:

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hayate@nanako+0:polyview$ scons
scons: Reading SConscript files ...
NameError: name 'Variables' is not defined:
  File "/home/hayate/archive/bin/polyview/SConstruct", line 20:
    vars = Variables()


Actually it's not that I insist on using new solutions... I don't have the older version of scons so I'm unaware when something no longer works on the old version. I'll see if I can fix this particular incompatibility, it shouldn't be too hard.

By "new solution", I was referring to scons as opposed to make.

If you've ever done anything non-trivial with make, you'll hate it just as much as I do.

Not too long ago I actually moved a project from using a .sh to using a Makefile. It allowed me to do lots of things I couldn't previously do. Such as having development and release builds, not having to rebuild every file when something was changed, and copying the static data directory from source to live copy (since the program can change its data and I'd rather that be reset to the original state when I rebuild, for example, in case a bug destroys it all). The only disadvantage I've found so far is that a few things have been a little hackish, but it still works fine.

scons, on the other hand, has consistently failed to work for me. No, I don't understand how scons works before, but I have tried many times to compile something with scons by following the readme file, and it comes up with some error or another that I don't have the knowledge to fix. Your program wasn't the first.

Why is it even necessary to clean the workspace in the first place?

Perhaps you actually want to get rid of all the temporary files the build script leaves lying around in a mess?

Shouldn't the build system know how to rebuild a project correctly without you having to babysit it every time?

If make does something that appears unnecessary or silly, that's not the fault of make, it's the fault of whoever wrote the Makefile.

Ever added a comment to a header file and have the entire project rebuild in 1.5 hours?

Why would you want to rebuild a project just because you added a comment?

Ever make a change and run 'make', and spend 6 hours debugging your program only to discover at the end that the code was correct, only make failed to rebuild a particular target?

I've wasted ludicrous amounts of time debugging non-bugs that were caused by using incorrect compiler settings in pretty much everything but make.

Worse yet, ever work over NFS and have make always rebuild the entire 2.5 million line codebase just because the NFS server's clock is slightly off from your workstation?

No, and that's silly anyway. If you're using NFS you should just be able to SSH into the server anyway and compile it there.

What's the point of a build system if you have to rebuild the whole damn thing every single time anyway? Why not just use a shell script of gcc commands?

I actually used to do this for a long time, and it wasn't particularly difficult.

Then you have people who use scripts and makefile hacks to coax it to do what they want, and the amount of makefile code exceeds the amount of source code?

At which point you may as well just write the build script in Python rather than bash...

And most of the time, these hacks are just houses of cards with global variables and whatnot that break subtly every time you try to do anything with it. Most of the time you don't notice the bugs until it comes back to bite you much later.

I could say that about normal program code too.

Make is at least 3 decades old and decrepit and is overdue for a long-deserved death.

C is nearly four decades old and it's still one of the best languages around.

[quickfur]

I extracted that over the same directory and I get the same error.

If I run scons in the polyview directory (instead of proglib), I get this:

Code: Select all

hayate@nanako+0:polyview$ scons
scons: Reading SConscript files ...
NameError: name 'Variables' is not defined:
  File "/home/hayate/archive/bin/polyview/SConstruct", line 20:
    vars = Variables()


Hmph. You are using a pre-release version of scons, you know that, right? I've tested on SCons 1.0.0 and everything builds fine. But no matter. I've made a different version of the SConscript specifically for you, so install the new sources here:

http://eusebeia.dyndns.org/polyview/polyview-3.0pre-3568.tar.gz

and instead of running 'scons', run 'scons -f SConscript.0.97'. (You only need to do this in the main directory; i've removed the offending rule from the proglib subdir because that file isn't even used anymore anyway, just leftover junk from long ago.)

[...] By "new solution", I was referring to scons as opposed to make.

Well, I'm not interested to start a flamewar on this, but it suffices to say that I found scons more suitable for what I need to do than make. Let's just leave it at that, ok?

If you've ever done anything non-trivial with make, you'll hate it just as much as I do.

Not too long ago I actually moved a project from using a .sh to using a Makefile. It allowed me to do lots of things I couldn't previously do.

Well, if you were using .sh before, then make certainly is a whole level up! I could say the same about make->scons, but I'll refrain.

[...] The only disadvantage I've found so far is that a few things have been a little hackish, but it still works fine.

Up to a certain point. Like I said, for simple projects it's good enough. But i work with very large and very complex projects in my job, and make is totally inadequate for that kind of thing. Not only things are hackish, but with so many people working on so many things at the same time, it becomes hack upon hacks upon hacks, and eventually it turns into a gigantic castle of cards that nobody dares touch lest the entire thing comes crashing down.

We did rewrite the makefiles at one point to clean things up, and things were (relatively) good for a while. But, inevitably, things just went on a downward spiral from there, and now we're about ready to rewrite the makefiles again. I can't speak for you, but I find this a total annoyance and a sign that something fundamental is wrong with the system. A proper build system shouldn't need you to rewrite your build scripts once every few years.

scons, on the other hand, has consistently failed to work for me. No, I don't understand how scons works before, but I have tried many times to compile something with scons by following the readme file, and it comes up with some error or another that I don't have the knowledge to fix. Your program wasn't the first.

Well, using a prerelease version doesn't help. But that aside, while I can't speak for others, the reason for failure in my program is because i've never actually tested it on anything other than my own system, so I've no idea when I did something to break it. And for the particular problem you're seeing, i've identified the source of the problem to be a new experimental feature that I used, which I didn't know wasn't part of the old version. The same kind of problems would crop up when, for example, people write makefiles using gmake-specific features.

Why is it even necessary to clean the workspace in the first place?

Perhaps you actually want to get rid of all the temporary files the build script leaves lying around in a mess?

'scons -c' takes care of that.

Whereas with make, you actually have to hand-code a clean rule. Which again, shows that as a build system it's rather lacking in basic features.

Shouldn't the build system know how to rebuild a project correctly without you having to babysit it every time?

If make does something that appears unnecessary or silly, that's not the fault of make, it's the fault of whoever wrote the Makefile.

Not necessarily. Make uses timestamps, which are unreliable. E.g., check out a file from your favorite VCS, and the timestamp gets updated, and make rebuilds the whole thing. Do this with the wrong file (e.g. a header that everyone includes) rebuilds the entire project.

Ever added a comment to a header file and have the entire project rebuild in 1.5 hours?

Why would you want to rebuild a project just because you added a comment?

What if you made a bunch of changes to other files as well? Technically speaking, only those changes should be rebuilt. But just because you happened to touch that header file that everyone uses, the entire project rebuilds. Whereas the other changes you made could've required only rebuilding 5% of the whole thing.

Ever make a change and run 'make', and spend 6 hours debugging your program only to discover at the end that the code was correct, only make failed to rebuild a particular target?

I've wasted ludicrous amounts of time debugging non-bugs that were caused by using incorrect compiler settings in pretty much everything but make.

Well I'm not talking about building something you downloaded, though. I'm talking about development. Perhaps you were lucky and worked on the right projects that you never had to deal with this kind of problem. I have to deal with this at work, every day. By now, I've given up on make ever doing the right thing. When something doesn't work, just delete the entire workspace, do a fresh svn checkout, and build from scratch. It wastes hours of work time every day. But I get paid to do this, so I shut up and deal with it.

Worse yet, ever work over NFS and have make always rebuild the entire 2.5 million line codebase just because the NFS server's clock is slightly off from your workstation?

No, and that's silly anyway. If you're using NFS you should just be able to SSH into the server anyway and compile it there.

This isn't always an option when you're not the only one working on the project. I certainly can't do this at my work, they'd kill me.

What's the point of a build system if you have to rebuild the whole damn thing every single time anyway? Why not just use a shell script of gcc commands?

I actually used to do this for a long time, and it wasn't particularly difficult.

Like I said, if the project is simple enough, this is all you need.

Then you have people who use scripts and makefile hacks to coax it to do what they want, and the amount of makefile code exceeds the amount of source code?

At which point you may as well just write the build script in Python rather than bash...

Well, yes. That's why i use scons.

And most of the time, these hacks are just houses of cards with global variables and whatnot that break subtly every time you try to do anything with it. Most of the time you don't notice the bugs until it comes back to bite you much later.

I could say that about normal program code too.

Ahhh... don't get me started on that. I see code written by others every day, and ... no, i'll shut up now.

Make is at least 3 decades old and decrepit and is overdue for a long-deserved death.

C is nearly four decades old and it's still one of the best languages around.

Because C is fundamentally correct in its design. Make has some really fundamental design issues at its core, that cannot be fixed without a radical redesign from ground up. It's about time people moved on. But whatever. I've already spent way too much time arguing about this. I'll shut up now.