IncMats Website Update

[Klitzing]

Hy folks,

I'd like to inform you, that my IncMats website is now updated once more.

Changes include:

• version history included
• download page added (incl. the there provided spreadsheets)
• lace cities moved to more prominent positions
  several of those ASCII-art pics with according true projection pics complemented
  several additional ones
• several additional lace towers incmats
• vertex layers according to subsymmetries added
• some own incmats files for general variations like a3b3c
• incmats-files with table-border switched on for more clarity at property separations
• prism-symmetric segmentochora added
• some youtube videos added
• extra page on axial polytopes added, also refering the EEAs and EEBs!
• segmentochoron pdf also updated (with footnotes)

and, for sure, lots of new or changed incmats pages...

Must have a look!
--- rk

[Klitzing]

Hy folks,

I'd like to inform you, that my IncMats website is now updated once more.

[...]

Must have a look!
--- rk

Just to inform you: Updated once more

Recent changes include e.g.

• started to collect convex segmentotera
• started to collect 4D CRFs, providing lots of recently found examples
• throughout incmats files: cross-links to hedrondude's and quickfur's websites (if applicable)
• links inserted in the version history
• started to group unsorted "confer"-lists now into contexts
• new page to collect noble polytopes
• extending the section on subsymmetrical diminishings
• inclusion of a section on Green's family of supersemicupolae to the axials page
• full table of convex segmentochora included
  and conversely added in incmats files the Klitzing numbers of segmentochora
• more than 100 additional incidence matrix files (with 90 therefrom for 4D)

Go, have a look!

--- rk

[Klitzing]

I'd like to inform you, that my IncMats website is now updated once more.

Just to inform you: Updated once more

Dear all,

just want you to inform on the yesterday update of my incmats Website
http://bendwavy.org/klitzing/home.htm


Some of the recent changes:

- new page on (partial) Stott expansion, together with many new corresponding incmats files of any geometry (curvature)
  http://bendwavy.org/klitzing/explain/stott.htm

- new page on dihedral angles
  http://bendwavy.org/klitzing/explain/dihedral.htm
- started to add dihedral angles within incmats-files

- incmats pages for J.McNeill's elementary honeycombs added, within those (and some applicable others too): cross-links to his website added
  http://bendwavy.org/klitzing/dimensions/flat.htm#elementary-honeycombs

- several new cross links to meanwhile added wiki pages
- incmats display optimized through javascript based content depending text-coloring (within ie only supported from ie9 on)
- several technical improvements on cross-browser support (incl. the new ie10)

- nearly 120 additional incidence matrix files


Have a look!

--- rk

[Klitzing]

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...)

Changes in detail:

• section on euclidean star tilings added
• lot more hyperbolic tesselations added (of various dimensions, both compact and non-compact)
• incmats-files for hyperbolic tilings with Coxeter domains added
• a listing added, providing now some more such 2D and even some 3D ones
• providing (purely imaginary) circumradii for hyperbolics too
• several further uniform polytera added, esp. those of hedrondude's "Polyteron of the day"
• some of the formers needed extra files for their intricate vertex figures
  (for calculating their circumradii a formula shows rather useful, which relates that of the original figure to that of its verf!)
• redesign of site map
• more than 150 new incidence matrix files
• more than 500 old ones got additions

--- rk

[student91]

Just read your page about dihedral angles, and I think you might be interested in my way of calculating dihedral angles. (Whereas your way is rather specific and doesn't need that much calculations, my way is easier appliable (you could use only incmaths), but involves more calculations.)

as dichoral angles are fixed for three cells meeting at an edge, I derived a formula for this case.
say you have three cells A,B and C meeting at an edge.
now the length of the base of an icosceles triangle with edges of length 1, and a top angle equal to the dihedral angle of A will be denotated a, so with B and C. (I used to calculate this by 2*sin([dihedral angle of A]/2), but sqrt( 2-2cos([dihedral angle of A]) ) also works, and is more favorable as much angles are given as arccos.

furthermore, i'll call 1-(b²/4)=b', and so with c
now the dihedral angle between B and C is given by:
arccos( 1/2*b/c*sqrt(c'/b') +1/2*c/b*sqrt(b'/c') -a²/(bc*sqrt(b'c')) ).

isn't that a neat formula?

student91

[Klitzing]

Sadly by mere short sight I cannot envision how you derived that ...
(Your a,b,c seem to be sides of some planar triangle with circumradius 1, b' and c' then would be respective inradii.)

Therefore I considered to derive a similar thing myself from scretch. Here is what I get:

An alternate derivation of dihedral angles is based on a localised and hierarchical access, which calculates angles across ridges (D-2 elements) between pairs of margins (D-1 elements) of some D-dimensional polytope around any threefold (and thus rigid) peak (D-3 element) of it. This is based on the spherical law of cosines (for sides) cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(gamma), which solves to gamma = arccos[(cos(c) - cos(a) cos(a)) / sin(a) sin(b)].

Here the above gamma would result in the searched for dihedral angle alpha_D of the D-dimensional polytope (at the considered ridge). And a,b,c represent the respective dihedral angles alpha_D-1 of the there incident D-1-dimensinal margin polytopes.

Esp. when all dihedral angles a = alpha_D-1 generally are provided by some algebraic expressions, i.e. cos(alpha_D-1) = [alg. expr.], terms like cos(a) are immediate. For the used terms sin(a) we'd thus get sin(a) = sin(arccos([alg. expr.])) = sqrt(1 - [alg. expr.]²). (Sign issues do not intervene here, at least as long as only dihedral angles alpha;_D-1 < pi will be involved. But this is just the domain of arccos. Therefore we will be safe here.)

Thus, if A,B,C are such algebraic expressions for cosines of incident dihedral angles alpha_D-1,
i.e. A = cos(a) etc., and still C will be the one opposite to the searched for dihedral angle alpha_D,
then we thus derive alpha_D = arccos[ (C - A B) / sqrt[(1 - A²)(1 - B²)] ].

In fact, this is (a bit more elaborate) what I did for 3D on my by you cited page, when considering rigid acrons.

--- rk

[faris]

I have been searching for any work done on a polychoron equivalent of the hypertorus, and I have found nothing.

[wendy]

You can get a hypertorus polychoron by multiplying a polygon by a polyhedron. It's called the comb product.

Klitzing's site is more about convex polychora, which means they are not "hypertoruses", what ever that might mean. Have a look at the torotope topic where many different torus-shaped things are discussed. 4d has three different kinds of toruses, at least. Maybe more.

[Klitzing]

Just want to inform you on a further update of my IncMats webpage.

- lots of further CRFs (esp. of the recently quite active CRFebruary ),
- several more polytera (accordingly to PolyhedronDude's recently rather quiet Polytera of the Day page )
- new page on closed finite flat complexes
- section on Coxeter domains extended
- section on ICN5D's |,>,O devices introduced
- more than 125 new incmats files   (with about 25 for 3D,   about 80 for 4D,   about 40 for 5D)

--- rk

[wendy]

In the 'closed finite flat complexes', the figures you list are 'piecewise finite', which means, that one can complete the full incidence of any surtope. There are other tilings, particularly in terms of the $\mathbb{Q}$ groups which are not piecewise finite. These include things like the tiling of tetrahedra in the euclidean plane.

You have of course, the o5o5/2oAo group in three dimensions. Another 4d group missing from your list is the one i designate 5BB, formed by two equal-sized o3o3oAoBo, in dual positions. It is infinitely dense, has six mirrors (at least) in its fundelental region, and corresponds to the tiling of octagonny, dual to the octagrammy. We could use your ø notation to see if we can come up with anything, though.

Your pages seems to get more activity than mine, although at the moment, i am rather engrossed in six-dimensional electronics.

[Marek14]

You have my name a bit wrong there -- It's supposed to be "M. Čtrnáct", not "Ctrnáct"

[Keiji]

Just want to inform you on a further update of my IncMats webpage.

Hooray - just in time for me to confirm my independently worked out imats for the J92 rhombochoron and its tetraaugmented variant

(also does this mean it's safe for me to start splitting up the Johnsonian Polytopes thread?)

$\mathbb{Q}$ groups

Is that supposed to say ℚ groups?

You have my name a bit wrong there -- It's supposed to be "M. Čtrnáct", not "Ctrnáct"

I'd really, really like to know how to pronounce that

[Klitzing]

You have my name a bit wrong there -- It's supposed to be "M. Čtrnáct", not "Ctrnáct"

Oops no harm intended. - Now corrected in my offline copy.
--- rk

[Klitzing]

(also does this mean it's safe for me to start splitting up the Johnsonian Polytopes thread?)

It ought to
--- rk

[Marek14]

I'd really, really like to know how to pronounce that

Well, you're in luck there, there's a link

http://www.locallingo.com/czech/phrases/numbers.html

And yes, my surname really IS a numeral. I wanted to become a mathematician with a name like that, but, sadly, it didn't work out

[wendy]

It sounds a good deal different to what the croat i worked with back in the seventies said of '4'. But then they're in different subgroups of the slavic family.

wendy wrote:$\mathbb{Q}$ groups

Too much dreaming in latex, i am afraid. Normally, i would not put the curlies in, but some folk don't pick up on that.

[Klitzing]

In the 'closed finite flat complexes', the figures you list are 'piecewise finite', which means, that one can complete the full incidence of any surtope. There are other tilings, particularly in terms of the $\mathbb{Q}$ groups which are not piecewise finite. These include things like the tiling of tetrahedra in the euclidean plane.

Well, that page is not on modular rings, i.e. on higher dimensional lattices being projected into some lower dimension. Rather it is about closed dyadic figures, which just happen to come out flat.

Sure one might use the formers to construct several of the latters. E.g. by combining 2 tesselations of identical patches, one being used as "nearer" and one as "farer" side. This is what is outlined there. But there are Independent approaches to such thingies as well. And some of the are outlined there too.

My listing surely is way from being exhaustive. Rather it is some collection only. But for sure, I'm willing to include further interesting ones as well.

--- rk

[Klitzing]

Dear polytopists,

just want you to inform about my recent update of my IncMats website.

The change history mentions:
- expanded kaleido-facetings extracted from (partial) Stott expansions
  resp. from CRFs into a new own subpage
- several new according EKF-CRFs have been found
- closed finite flat complexes largly extended and lots of according incmats added
- major revision of section on Partial Stott Expansions
- lots of non-convex polytera incidence matrix files added
  (thanks to the recently updated listing resp. pictorial groundworks of HedronDude!)
- all irreducible convex Wythoffian polyzetta now having acronyms
- noble page now including the (so far elaborated) euclidean and hyperbolic ones
- opened the challange for compounds beyond 4D as well
- due to EKF research several more subsymmetrical representations of the hexacosachoron (ex) have been added
- because of several quite huge incidence matrices in a single file the soften script for their zeroes might run rather long;
  therefore an interactive skipping possibility now has been added to some of the most extremal cases
  (both, soften script and its skipping are available only if javascript execution is being allowed)
- tegum sum defined and contrasted to the tegum product,
  incl. providing its relation to the degenerate zero-height lace prisms with pseudo bases
- several further incidence matrices according to tegum sum representations added
- neerly 150 new incidence matrix files / neerly 800 edited ones

--- rk

[Klitzing]

Today the next release of my incmats webpage was uploaded.

- more than 100 new incmats files
- large increase on decompositions
- & many things more, cf. version history

@keiji: cross-links into this forum now all refer by corrected new URL.

--- rk

[Marek14]

- recent research results on Some Hyperbolic Tiling Classes with Specific Local Configuration Types added

Can't seem to open that page

[Klitzing]

Guess what? -
 - Today the next release of my IncMats webpage was uploaded.

For details on changes cf. version history.


Also several things discussed in this forum in that last period are now detailed therein. E.g. the gyrotrigonisms of this thread, or the Dutour polytopes of this thread.

But I also started there to write on quasicrystals & quasiperiodicity, elaborating on different projection methods. In this context I set up a new subpage on lattices as well. And even an own subpage on the Gosset polytopes was piled up for a quicker, cross-dimensional access.

Both activities resulted not only in several new polytopal IncMats-files, but also in quite huge further subsymmetrical IncMats-descriptions of lots of older files too. - So, take a look!

--- rk

[Klitzing]

Want to inform you all about a further IncMats website update.

Updates include:

• more than 100 new IncMats files
• a separate page on the dimensional behaviour of convex unit-edged lace prisms (providing lots of explicite examples)
• a separate page on scaliform polytopes (providing examples from 4D to 7D)
• a downloadable excel spreadsheet listing the known 4D CRFs by their different cell counts

--- rk

[Klitzing]

It looks to be rather quiet within this forum, lately...

So I’d like to inform you about a further update of my IncMats website at http://bendwavy.org/klitzing/home.htm

Updates include:

• started to collect convex segmentoexa (i.e. 7D monostratic orbiform convex polytopes)
• separate page on skew polytopes
• a small subsection on skeletal polytopes added
• added a further page for asorted other polytopes – meant to host temporarily so far otherwise nowhere else linked polytopes
  (e.g. some polyhedra, polychora, or honeycombs, which incorporate rhombs, shields, and other non-regular unit-edged faces;
  perfect Gévay polychora;
  some combinatorically regular polyhedra of higher genus)
• more than 80 new incmats files, more than 120 new incidence matices
• esp. matrices for 2 further bi-cyclopentadiminished figures: bicypdrox, bicypdrahi (in addition to bicypdex)
  (bicypdrahi had been discussed originally within this forum here, here, and here - way back in Februrary 2013,
  while bicypdrox here and here - in January 2014)

Just go and have a look!

--- rk

[Klitzing]

[...]

• added a further page for asorted other polytopes – meant to host temporarily ...
    (e.g. ...
    perfect Gévay polychora;
    ...)

[...]
--- rk

Got those Gévay polytopes from his according publication "Construction of Non-Wythoffian Perfect 4-Polytopes", Gábor Gévay, Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Volume 44 (2003), No. 1, 235-244. - Or online accessible here.

Therein he makes a longuish construction of perfect polychoroa which by construction are non-Wythoffian. The construction then starts with some regular polychoron {P,Q,R} and inserts into each of its cells {P,Q} a symmetrically matching oPmQo (= dual of rectified {P,Q}), such that one type of its vertices coincides with the face centers of the starting polychoron, while the other vertex type lies internal to those original cells. Next he applies a 4D convex hull operation to just those inscribed cells, which then results in these polytopes.

But now it appeared to me that these polychora could be described much easier!
They simply are   r ( d ( r ( {P,Q,R} ))),   where r means the rectification operator and d means the dualisation operator.
Or, by using again Wendy's mirror-margin edge notation, they are just
r ( d ( r ( xPoQoRo ))) = r ( d ( oPxQoRo )) = r ( oPmQoRo ).

--- rk

[Klitzing]

In a private conversation with Gábor Gévay, he reminds me that the rectification operation is well-defined for uniform starting polytopes, as one just is allowed to use the edge midpoints. But for multiform polytopes, i.e. ones with different vertex classes, this might not be the case. Quite generally the depth of truncation of the hyperplanes on all vertices of a single class surely is the same. Accordingly edges between 2 such alike vertices would then provide again a vertex at its midpoint. But the according depth at vertices of different classes need not be related at the first sight.

But at least wrt. oPmQoRo we are quite lucky. Cause a truncation (and thence a rectification) of that polychoron implies too a truncation of its cells. And in this specific cases those are all bipyramids co oxRoo&#zy = oRo || xRo || oRo (where c is just the total height from one tip to the other and y is the according lacing edge size, while x is its equatorial one).

Let's consider the truncational planes wrt. the equatorial vertices. Those provide rhombs of varying sizes (according to the truncational depth). Along the equatorial edge of the starting bipyramid the special truncational depth, asked for rectification, clearly is uniquely defined, here resulting in the midpoints of the original equatorial edges. But this then also settles the searched for points on the lacing edges as well (just by application of these rhombs). Therefore the required depth of truncation for the 2 tips thus is given, when searching for a rectification, i.e. the coincidence of the truncational vertex pairs on all edges.

Therefore rectification Comes out to be well-defined here too.
In fact, Gévay's attribution of "perfect polytopes" after all is nothing but the well-definedness of the (outermost) rectification operation in   r ( d ( r {P,Q,R} ))) = r ( oPmQoRo ).

--- rk

[polychoronlover]

I noticed that the IncMats website says that o3o3x3x3/2 *a and o3x3x3o3/2 *a lead to 2 superimposed copies of the facetorectified tesseract, firt. However, due to the possibility of demitesseractic symmetry, and the fact that the vertex figure can reduce its symmetry from trigonal prismatic to trigonal pyramidal (see sto and gotto), it appears that o3o3x3x3/2 *a and o3x3x3o3/2 *a form firt, not 2firt.

[username5243]

You might be right.

I've been thinking of another type of degenerate polychoron here. For instance, on 3D he has cho listed as a reduced version of x3/2x3x, removing the degenerate x3/2x faces. So I wondered if there was an analogous thing in 4D.

So I started with x3/2x3x3x. This thing has as cells 5 x3/2x3x's, 5 toes (x3x3x), 10 x3/2x x's (basically a double covered trip), and 10 hips. Then I wondered what would happen if the "2trip"s were removed, and the x3/2x3x were reduced to choes. It would have cells then as 5 choes, 5 toes, 10 hips.

A quick search on Hedrondude's website revealed that the result is the polychoron called ripdip (see Category 10, in Prip regiment). And that means x3/2x3x3o also has a similar reduced case, where removing the x3/2x faces makes a sirdop (see Category 6, in srip regiment).

I'm very interested to find some more similar polychora to these ones. First off, there is probably an analogue to x3/2x3x3x in highr symmetries, like x3/2x3x4x or x3/2x3x5x.

Anyone has any further ideas?

[polychoronlover]

What stopped me from looking into this is that there are multiple ways to represent each shape degenerately by a diagram with xn/2x. For example, ripdip can be expressed as x3/2x3x3x - 20*x3/2x, but also as x3x3x3x3/2 *b - 20*x3/2x. One of the main reasons I chose to look at truly Whythoffian polychora, and even hemi-Whythoffian ones, was that there was always only one symbol for the shape that could be derived from the vertex figure (apart from the equivalent symbols that could be found by replacing n by n/(n-1) on certain branches, but even then there was usually one case that resulted in the most uniform polychora). Another reason was that I had only ever gotten practice grouping nonconvex polychora by regiment. Grouping them by Dynkin diagram found a lot of associations between polychora that I had previously thought were unrelated. I wonder how many Whythoffian polychora can be reached by starting with one, searching its family, searching each family members' regiments, searching the regiment members' families, etc.

But I agree, this looks like a big untapped source of new ways to make familiar polytopes. I might look into it some time.

[username5243]

Indeed.

Even in 3-D, some polyhedra have two representations - for instance, sird (in srid regiment) can be given as x5/2x5x - 12{10/b}, but also x3/2x5x - 20{6/2}.

You know what? I'm going to start a new thread to do a full investigation into these polytopes. I'll probably start out with the linear o3o3o3o and all variants from it with 1 or more 3s changed to 3/2.

[polychoronlover]

For what it's worth, not every possible set of x -> x/(x - 1) substitutions on a Dynkin diagram even leads to a figure with the same symmetry. For example, o3o3o3/2 *a and o3/2o3/2o3/2 *a both have tetrahedral symmetry, but o3o3o3 *a has triangular tiling symmetry, and o3o3/2o3/2 *a has either triangular tiling or triangular pyramidal symmetry, depending on how you close it. (However, it happens that for o3o3o3o, every possible substitution works.)

It turns out that for oAoBoC *a, the three substitutions that are guaranteed to work are to replace A, B, and C by:

• A/(A - 1), B/(B - 1), C
• A/(A - 1), B, C/(C - 1)
• A, B/(B - 1), C/(C - 1)

For the general tetrahedral diagram, oAoBoCoD *aEc *bFd, the seven working substitutions are to replace A, B, C, D, E, and F by:

• A/(A - 1), B, C, D/(D - 1), E/(E - 1), F
• A/(A - 1), B/(B - 1), C, D, E, F/(F - 1)
• A, B/(B - 1), C/(C - 1), D, E/(E - 1), F
• A, B, C/(C - 1), D/(D - 1), E, F/(F - 1)
• A/(A - 1), B/(B - 1), C/(C - 1), D/(D - 1), E, F
• A/(A - 1), B, C/(C - 1), D, E/(E - 1), F/(F - 1)
• A, B/(B - 1), C, D/(D - 1), E/(E - 1), F/(F - 1)