IncMats Website Update

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Re: IncMats Website Update

Postby Marek14 » Mon Jun 24, 2019 9:20 am

Klitzing wrote:Just 2 months later the next update of my IncMats website is up already! 8)

2019 / 6 / 22
- added an own page on ridge facetings as well as several according sub-pages
- added an own page on lace simplices, listing all the easiest cases each
- added a section on the Dehn-Summerville equations for simplicial polytopes
- several broken links, esp. on the pages for hyperbolic tilings and to pics of polyterons, now got fixed
- exactly 50 new incmats files as well as nearly 150 additional incidence matrices, many of them for various segmentotera, thus summing up to the following totals:

Total count of actually available polytopes: 3448
Total count of contained incidence matrices: 8256

--- rk


Speaking of hyperbolic tilings, I made some new hi-def images of some, thanks to new HyperRogue features that allow for displaying of general Archimedean tilings. If you want, we could update their page :)
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Re: IncMats Website Update

Postby Klitzing » Sat Nov 23, 2019 9:06 pm

Next IncMats website update got uploaded.

Changes 2019 / 11 / 23:

- completed the list of all possible up to 5D lace simplices, which use links marked 2 or 3 only
- further downloadable excel spreadsheets, eg. in order to calculate the circumradius of lace prisms and of trigonics respectively
- several new uniform compounds, some with ex components:
  spohi (5) and sody (10), kepisna (6) and pedisna (12), kidisna (10) and ditusna (20);
  others with ico components: kitapna (4) and bitapna (8), kitefa (6) and bitefa (12)
- added a section on the Coxeter-Moser polytopes, a.k.a. regular maps
- even more broken links, esp. to pics of polyterons, now got fixed
- cross-linking to Nan Ma's page on regular stars
- nearly 60 new incmats files as well as nearly 150 additional incidence matrices, many of them in 5D


(Sorry @Marek14 for not having replied to you so far. I did get your provided pics. But still hadn't enough time to exchange them one by one.)

--- rk
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Re: IncMats Website Update

Postby Klitzing » Wed Jul 01, 2020 10:01 am

Hi,

just want to inform you about the recent update of my incmats website http://bendwavy.org/klitzing/home.htm .
(In case don't miss to hit the reload button to get the newed pages and not those from your cache...)

Some of the recent changes:

- many more 4D uniform compounds listed, even started to collect 7D & 8D compounds
- also extended to 7D combinations of extrusions and taperings, cf. |,>,O devices
- the notion of alterprisms is being added (thereby elaborating various further scaliform examples),
  as well as introducing according acronym shortenings (X-al-X becoming now simply X-a)
- some more polychora investigated wrt. EKF constructions therefrom
- hidicau pretasto, a further outstanding CRF has been found after a long time of silence in this resort
- section on the hypersine function for hypersolid corner angles
- cross-linking to a new polytopewiki
- over 80 new incmats files as well as 160 additional incidence matrices

Have a look!

--- rk
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Re: IncMats Website Update

Postby Klitzing » Tue Dec 22, 2020 9:15 am

Hi,

just want to inform you about the recent update of my incmats website http://bendwavy.org/klitzing/home.htm .
(In case don't miss to hit the reload button to get the newed pages and not those from your cache...)

Some of the recent changes:

- added a page on isogonal polytopes with quite a lot of according stuff from a current research project
- noticed that firp in fact is a 4D cuploid and found a further such: co retro-cuploid
- found a new scaliform honeycomb with vertex configuration 5 Y4 + 3 T + 3 Q3 + 1 T3
- continued cross-linking of the still quite fast growing polytopewiki
- started to collect some twins
- major xhtml-reworking: erasing lots of html-allowed, but not well-defined xml
- finally providing OBSAs for the polygons as well
- started to provide OBSAs for hyperbolics as well
- started to collect compounds of euclidean tilings as well
- nearly 200 new incmats files as well as more than 250 additional incidence matrices

Have a look!

--- rk
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Re: IncMats Website Update

Postby Marek14 » Tue Dec 22, 2020 9:18 am

Me and Zeno have been collecting my tilings at https://zenorogue.github.io/tes-catalog/ . I started in tilings many years ago by researching uniform (a,a,a,b) tilings, and now I have finished the classification of 2-uniform (a,a,a,b) as well :)
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Re: IncMats Website Update

Postby Klitzing » Tue May 04, 2021 10:57 am

Just want to announce that even before Jonathan' polychoron website
my own IncMat website https://bendwavy.org/klitzing/home.htm has got an update!
Go and check out! - (You might want to hit [F5] if the cache displays the old content still.)

Excerpts from the new stuff:
  • hidlin and idinaq, 2 newly found 7D convex scaliforms
  • sissiddow, a 7D non-convex noble scaliform
  • quite a few newly found 6D convex scaliforms
  • major rewrite and extension of skew page while finding infinite skew polychora
  • further isogonal doubling cases elaborated
  • started to collect non-convex non-Wythoffian polypeta, polyexa, and polyzetta
  • started with isogonal polytera
  • elaborated a further 3D lace hyper city for the 8D 2_4,1
  • starting to provide a small intro on Shephard-Coxeter Polytopes (aka complex polytopes),
    giving additionally their relations to the according twice-dimensional real space polytope,
    from which it is just the abstract polytopal subset of (some of) its even-dimensional elements
  • reached the limit of 4000 html files in the whole IncMats website,
    providing more than 100 new polytope files and above 175 additional incidence matrices
--- rk
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Re: IncMats Website Update

Postby mr_e_man » Fri May 21, 2021 4:33 am

Lots of good information there. :) I didn't know what the Gosset polytopes km,n were until I found it on your site.

Do you think any of my results on CRF honeycombs are worthy to be mentioned there? They can be summarized thus: Any CRF honeycomb, other than cube-doe-bilbiro, has only cells which can be decomposed into P3,P4,P8,T,Y4,Q4 . In particular, decagons cannot be used, for example.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
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Re: IncMats Website Update

Postby Klitzing » Sat May 22, 2021 5:46 pm

mr_e_man wrote:Do you think any of my results on CRF honeycombs are worthy to be mentioned there? They can be summarized thus: Any CRF honeycomb, other than cube-doe-bilbiro, has only cells which can be decomposed into P3,P4,P8,T,Y4,Q4 . In particular, decagons cannot be used, for example.

Is that statement of yours a mere observation based on the known ones,
or is it a rigorously proven result, valide also for any future being found honeycomb?
--- rk
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Re: IncMats Website Update

Postby mr_e_man » Sun May 23, 2021 7:59 pm

The latter: I have proven to my own satisfaction that other honeycombs cannot possibly exist.

In other words, the statement is valid for any honeycomb found in the future.

That linked page is supposed to be an outline of the proof. Anyone who wants to follow what I did there can point out any unclear parts that I should explain, or any cases that I missed.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
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Re: IncMats Website Update

Postby mr_e_man » Fri Jun 11, 2021 2:58 am

How should I respond to no response? :sweatdrop:

Is this result just not very interesting? Or is my "proof" totally incomprehensible? Or are you taking your time to figure it out?
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
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Re: IncMats Website Update

Postby Klitzing » Fri Jun 11, 2021 9:49 pm

When I read your starting article here I can get the followings

  • 2D convex tilings with regular polygons only can use 3-, 4-, 6-, 8-, and 12-gons,
    as any other being used regular convex polygon ultimatly results in a non-continuable configuration
  • morover there is only a single 2D convex tiling with regular polygons only using 8-gons: the uniform 4.8.8-tiling
  • occurances of 6-gons variously might be decomposed into 6 3-gons each (6-pyramid)
  • occurances of 12-gons variously might be decomposed into 6 3-gons, 6 4-gons, and 1 6-gon each (6-cupola),
    in fact within 2 different orientations.
  • 3D convex tilings with regular polygons should similarily be restricted to use 3-, 4-, 5-, 6-, 8-, 10-, and 12-gons only,
    as any other being used regular convex polygon ultimatly results in a non-continuable configuration
  • morover usages of 12-gons only can occur within infinite stacks of according 2D tilings.
  • sevaral CRF cells can also be excluded for similar reasons:
    4-ap, n-ap with n>5, 7-p, 9-p, 10-p, 11-p, n-p with n>12, J52-53, J84-90, J92, snic, snid, grid
About further restrictions on the usage of 10-gons - at least therein - you still seem unfixed.
Within a later post of that very thread you show up a configuration with a tid, but I don't get exactly why it isn't continuable.
Still, that one alone doesn't rule out any other 10-gon usage.

A bit later in that thread you further observe the (non-new) facts of degenerate segmentochora
  • tic || cube
  • toe || oct
  • co || point
  • girco || sirco
as well as the obvious decomposition of sirco into the lace tower squacu || op || squacu.

From that you now deduce that ANY 3D convex tiling with regular polygons should be decomposable into P3 (trip), P4 (cube), P8 (op), T (tet), Y4 (squippy), and Q4 (squacu)
- except for the single case of Weimholt's cube-doe-bilbiro honeycomb.

First of all I don't see the final 10-gon exclusion. But esp. I don't see why that mentioned exception should be the only possible 5-gon usage.

--- rk
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Re: IncMats Website Update

Postby mr_e_man » Sun Jun 13, 2021 10:44 pm

Let's discuss this over there, and not clutter your Update page too much. :arrow:
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Re: IncMats Website Update

Postby Klitzing » Thu Sep 09, 2021 2:12 pm

Just want to announce that the IncMat website https://bendwavy.org/klitzing/home.htm has got a further update!
Go and check out! - (You might want to hit [F5] if the cache displays the old content still.)

Excerpts from the new stuff:
  • worked out the lace city of riffy under A2×E6 subsymmetry
  • setup of an own page on dimensional analogs, providing various counts and measures in a general form
  • accordingly filled in the missing infos for the there contained polytopes in their individual pages as well
  • started for hyperbolic compounds
  • added further complex polytopes
  • above 120 additional incidence matrices
--- rk
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Re: IncMats Website Update

Postby mr_e_man » Wed Dec 01, 2021 5:56 pm

I noticed that incidence matrices could be made more consistent by including the "improper" faces, and letting the diagonal blocks be identity matrices. The introductory example you give, a tetrahedron with digonal antiprism symmetry, could be described by this incmat:

Code: Select all
1 | 4 | 2 4 | 4 | 1  nulloid
--+---+-----+---+---
1 | 1 | 1 2 | 3 | 1  vertices
--+---+-----+---+---
1 | 2 | 1 0 | 2 | 1  horizontal edges
1 | 1 | 0 1 | 2 | 1  diagonal edges
--+---+-----+---+---
1 | 3 | 1 2 | 1 | 1  faces
--+---+-----+---+---
1 | 4 | 2 4 | 4 | 1  body

Sure, the other version is more compact, with two fewer rows & columns.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
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Re: IncMats Website Update

Postby mr_e_man » Wed Dec 01, 2021 7:07 pm

Have you thought about a metrical extension to incidence matrices? I mean a complete description of the polytope's geometry, not just the abstract structure.

Obviously this could be done by giving coordinates for the vertices (possibly in a skewed system as with Wendy's notation), in addition to the incmat itself (as the vertices alone aren't enough to describe a non-convex polytope).

Instead of coordinates, I'm considering the measures of all dyads in the polytope. A k-dyad is a (k-1)-face and a (k+1)-face and two k-faces incident with them; this is abstractly the line segment, the 1-polytope. So the measures are edge lengths, vertex angles in a polygon, dihedral angles in a polyhedron, dichoral angles in a polychoron, and so on.

If we want to describe the geometry of non-convex polytopes, unsigned angles (in the interval (0°, 180°)) are not sufficient, as shown by these two heptagons:

Code: Select all
         o
        / \
       /   \
o-----o     o-----o
|                 |
|                 |
|                 |
|                 |
o-----------------o


o-----o     o-----o
|      \   /      |
|       \ /       |
|        o        |
|                 |
o-----------------o

They have the same edge lengths, and the same unsigned angles: 90°, 90°, 90°, 90°, 120°, 60°, 120°. To distinguish them we need signed angles: the first polygon has 90°, 90°, 90°, 90°, -120°=240°, 60°, -120°; and the second one has 90°, 90°, 90°, 90°, 120°, -60°, 120°. In fact this describes an insided polytope. A polygon can be turned inside-out by negating all angles.

A generic tetrahedron is partially described by this incmat:

Code: Select all
1 | 1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1 | 1   nulloid
--+---------+-------------+---------+---
1 | 1 0 0 0 | 1 1 0 1 0 0 | 1 1 1 0 | 1   vertex 1
1 | 0 1 0 0 | 1 0 1 0 1 0 | 1 1 0 1 | 1   vertex 2
1 | 0 0 1 0 | 0 1 1 0 0 1 | 1 0 1 1 | 1   vertex 3
1 | 0 0 0 1 | 0 0 0 1 1 1 | 0 1 1 1 | 1   vertex 4
--+---------+-------------+---------+---
1 | 1 1 0 0 | 1 0 0 0 0 0 | 1 1 0 0 | 1   edge 12
1 | 1 0 1 0 | 0 1 0 0 0 0 | 1 0 1 0 | 1   edge 13
1 | 0 1 1 0 | 0 0 1 0 0 0 | 1 0 0 1 | 1   edge 23
1 | 1 0 0 1 | 0 0 0 1 0 0 | 0 1 1 0 | 1   edge 14
1 | 0 1 0 1 | 0 0 0 0 1 0 | 0 1 0 1 | 1   edge 24
1 | 0 0 1 1 | 0 0 0 0 0 1 | 0 0 1 1 | 1   edge 34
--+---------+-------------+---------+---
1 | 1 1 1 0 | 1 1 1 0 0 0 | 1 0 0 0 | 1   face 123
1 | 1 1 0 1 | 1 0 0 1 1 0 | 0 1 0 0 | 1   face 124
1 | 1 0 1 1 | 0 1 0 1 0 1 | 0 0 1 0 | 1   face 134
1 | 0 1 1 1 | 0 0 1 0 1 1 | 0 0 0 1 | 1   face 234
--+---------+-------------+---------+---
1 | 1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1 | 1   body

We also need to specify the edge lengths l(i)(j), face-vertex angles a(i)(j)(k) [at vertex (i)], and dihedral angles h(i)(j). (Of course the edge lengths alone determine the angles if the geometry is known to be Euclidean or hyperbolic etc.) We could put these in another matrix, two blocks above and below the diagonal blocks, wherever there's a '1' in the incidence matrix:

Code: Select all
    |                     | l12 l13 l23 l14 l24 l34 |                     |       nulloid
----+---------------------+-------------------------+---------------------+-----
    |                     |                         | a123 a124 a134      |       vertex 1
    |                     |                         | a213 a214      a234 |       vertex 2
    |                     |                         | a312      a314 a324 |       vertex 3
    |                     |                         |      a412 a413 a423 |       vertex 4
----+---------------------+-------------------------+---------------------+-----
l12 |                     |                         |                     | h12   edge 12
l13 |                     |                         |                     | h13   edge 13
l23 |                     |                         |                     | h23   edge 23
l14 |                     |                         |                     | h14   edge 14
l24 |                     |                         |                     | h24   edge 24
l34 |                     |                         |                     | h34   edge 34
----+---------------------+-------------------------+---------------------+-----
    | a123 a213 a312      |                         |                     |       face 123
    | a124 a214      a412 |                         |                     |       face 124
    | a134      a314 a413 |                         |                     |       face 134
    |      a234 a324 a423 |                         |                     |       face 234
----+---------------------+-------------------------+---------------------+-----
    |                     | h12 h13 h23 h14 h24 h34 |                     |       body

This is clearly unsatisfactory for several reasons. The matrix is mostly wasted space. If the lengths and angles are given numerically with many digits, then the matrix becomes much wider and wastes more space. This notation doesn't work for the symmetry-specialized type of incmat; in the digonal antiprism, a vertex is incident with 3 equivalent triangular faces, but the 3 angles are not all the same, so we can't put a single angle in place of the '3'.

Any ideas for a better notation, which still shows all the lengths and angles explicitly?
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Re: IncMats Website Update

Postby Klitzing » Wed Dec 01, 2021 9:08 pm

mr_e_man wrote:I noticed that incidence matrices could be made more consistent by including the "improper" faces, and letting the diagonal blocks be identity matrices. The introductory example you give, a tetrahedron with digonal antiprism symmetry, could be described by this incmat:

Code: Select all
1 | 4 | 2 4 | 4 | 1  nulloid
--+---+-----+---+---
1 | 1 | 1 2 | 3 | 1  vertices
--+---+-----+---+---
1 | 2 | 1 0 | 2 | 1  horizontal edges
1 | 1 | 0 1 | 2 | 1  diagonal edges
--+---+-----+---+---
1 | 3 | 1 2 | 1 | 1  faces
--+---+-----+---+---
1 | 4 | 2 4 | 4 | 1  body


Sure, the other version is more compact, with two fewer rows & columns.


Nice find / rewrite. - But then my form of the matrices, i.e. providing the respective total element counts on the diagonal, have the additional nice relation on any diagonal "square":
Code: Select all
Mii * Mij = Mji * Mjj (for i<j)
which yours really misses - or at least would be burried somehow awkward elsewhere.

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Re: IncMats Website Update

Postby Klitzing » Wed Dec 01, 2021 9:26 pm

mr_e_man wrote:[...]Any ideas for a better notation, which still shows all the lengths and angles explicitly?


Yes.

In fact I sometimes (whenever it describes polytopes with different edge sizes) provide the corresponding edge size right behind the resp. edge type row.
The dihedral (ditopal) angles usually are given summarized at a different page position, but in fact are directly connected to the ridge type rows, i.e. could be given directly thereafter likewise.

Thus, examplifying at an "obvious" example: the tesseract could be given as:
Code: Select all
o3o3o4x

. . . . | 16 |  4 |  6 | 4
--------+----+----+----+--
. . . x |  2 | 32 |  3 | 3   (edge size =) 1
--------+----+----+----+--
. . o4x |  4 |  4 | 24 | 2   (dihedral angle =) 90°
--------+----+----+----+--
. o3o4x |  8 | 12 |  6 | 8

(Same applies with different edge types as well as different ridge types.)

--- rk
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Re: IncMats Website Update

Postby mr_e_man » Thu Dec 02, 2021 3:36 am

Klitzing wrote:
mr_e_man wrote:[...]Any ideas for a better notation, which still shows all the lengths and angles explicitly?


Yes.

In fact I sometimes (whenever it describes polytopes with different edge sizes) provide the corresponding edge size right behind the resp. edge type row.
The dihedral (ditopal) angles usually are given summarized at a different page position, but in fact are directly connected to the ridge type rows, i.e. could be given directly thereafter likewise.

Then you're only describing the 0-dyads and (N-1)-dyads. (Of course N is the dimension.) I also want the measures of the in-between dyads.

For example, the 16-cell (x3o3o4o) has 1, 60°, 70.5288°, 120°, as its dyad measures.
Last edited by mr_e_man on Thu Dec 02, 2021 4:54 am, edited 1 time in total.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
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Re: IncMats Website Update

Postby mr_e_man » Thu Dec 02, 2021 4:14 am

Klitzing wrote:
mr_e_man wrote:I noticed that incidence matrices could be made more consistent by including the "improper" faces, and letting the diagonal blocks be identity matrices. The introductory example you give, a tetrahedron with digonal antiprism symmetry, could be described by this incmat:

Code: Select all
1 | 4 | 2 4 | 4 | 1  nulloid
--+---+-----+---+---
1 | 1 | 1 2 | 3 | 1  vertices
--+---+-----+---+---
1 | 2 | 1 0 | 2 | 1  horizontal edges
1 | 1 | 0 1 | 2 | 1  diagonal edges
--+---+-----+---+---
1 | 3 | 1 2 | 1 | 1  faces
--+---+-----+---+---
1 | 4 | 2 4 | 4 | 1  body


Sure, the other version is more compact, with two fewer rows & columns.


Nice find / rewrite. - But then my form of the matrices, i.e. providing the respective total element counts on the diagonal, have the additional nice relation on any diagonal "square":
Code: Select all
Mii * Mij = Mji * Mjj (for i<j)
which yours really misses - or at least would be burried somehow awkward elsewhere.

--- rk

That becomes M(0,i) * M(i,j) = M(0,j) * M(j,i) (where '0' labels the nulloid). Both sides of the equation are counting the total number of incident pairs of 'i' and 'j' type elements; that is the total number of incident triples of 'i' and 'j' and '0' type elements.

Generalizing, how can we count the total number of incident triples of 'i', 'j', and 'k' type elements? The result should be invariant under permutations of i,j,k, which would give us some more equations. It's not simply M(0,i) * M(i,j) * M(j,k) ....
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Re: IncMats Website Update

Postby mr_e_man » Thu Dec 02, 2021 9:29 pm

Anyway, that modification of the incmat isn't important. You keep using yours, I'll keep using mine, it's easy to convert between them, it doesn't matter.

Let's look at the digonal antiprism again (omitting the digon faces, as usual). Suppose it has two edges with length 1 and four edges with length 2. The triangular faces are isosceles, with two angles arccos(1/4)=75.52° and one angle arccos(7/8)=28.96°.

Code: Select all
1 | 2 1 | 1 2 | 1  nulloid
--+-----+-----+---
1 | 1 0 | 1 1 | 1  base vertices; angle 75.5225°
1 | 0 1 | 0 2 | 1  apex vertex; angle 28.9550°
--+-----+-----+---
1 | 2 0 | 1 0 | 1  base edge; length 1
1 | 1 1 | 0 1 | 1  slant edges; length 2
--+-----+-----+---
1 | 2 1 | 1 2 | 1  triangle body

Code: Select all
1 | 4 | 2 4 | 4 | 1  nulloid
--+---+-----+---+---
1 | 1 | 1 2 | 3 | 1  vertices
--+---+-----+---+---
1 | 2 | 1 0 | 2 | 1  horizontal edges; length 1, dihedral angle 29.9264°
1 | 1 | 0 1 | 2 | 1  diagonal edges; length 2, dihedral angle 86.1774°
--+---+-----+---+---
1 | 3 | 1 2 | 1 | 1  faces
--+---+-----+---+---
1 | 4 | 2 4 | 4 | 1  tetrahedron body

How can we combine these lengths and angles (1, 2, 28.96°, 29.93°, 75.52°, 86.18°) into a single data structure?
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
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Re: IncMats Website Update

Postby Klitzing » Sat Dec 04, 2021 10:13 pm

you could split the vertices and the faces accordingly in order to include the full matrix of the faces into the matrix of the disphenoid:
Code: Select all
2 * | 1 2 0 | 2 1
* 2 | 0 2 1 | 1 2
----+-------+----
2 0 | 1 * * | 2 0
1 1 | * 4 * | 1 1
0 2 | * * 1 | 0 2
----+-------+----
2 1 | 1 2 0 | 2 *
1 2 | 0 2 1 | * 2

--- rk
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