User:Dc.samizdat/sandbox

Think

 * In Russia, the Kremlin reads what you write on Facebook. In America, the Kremlin writes what you read on Facebook.

Vector 2022 TOC
There are many things about the new 2022 interface that made me a bit uncomfortable on first using it, but in my experience so far the designers have made only one game changer, deal breaker change, by removing a feature I can't give up, so I will stay with the 2010 interface forever if they don't bring that feature back, at least as a user appearance preference. That's their removal of the old inline TOC at the top of the article, of course. The new pop-up sidebar TOC with its floating button is not a static TOC, it's a different feature entirely, innovative and useful in its own way (although the way its floating button always blocks the upper left corner of the page is very visually annoying, and you cannot get it out of the way no matter what you do by repositioning the page). But no matter -- that's not the deal-breaker. The pop-up sidebar TOC, whether you like it or not, isn't a TOC at the beginning of the article, which has been the signature appearance of every Wikipedia article since time immemorial.

When you open a Wikipedia article you expect to see a lede (like the abstract of a research article), followed by a table of contents showing the structure and organization of the article, giving you an instant idea of whether this article is 1 or 100 pages long, and how developed it is. As you refer to the article again and again over time, you will probably depart from that TOC to places you have discovered within the article again and again, your body developing a kind of muscle memory for the way the space inside the article branches out from the top. Your mind is learning the geometry of part of the vast space that is Wikipedia. The TOC at the top of every article illustrates one local part of that space. The TOC is the article editors' best attempt to choose a geometry for that subject that makes sense. It is editor-written content, artistry, not merely a generated index or search results; in fact it is the most important content in the article, after the lede. Sometimes it's all you read of an article (the lede and the TOC), and it tells you that you don't need to know any more. It can be collapsed or expanded, as suits your personal need of it, but surely it should not be entirely hidden in an always-collapsed pop-up sidebar.

The designers should fix this flaw in the new interface by simply bringing back the static TOC exactly as it is in the 2010 interface. The pop-up sidebar displaying the TOC can remain too, just don't display its floating button until the display is scrolled down to below the static TOC. It would also be a diplomatic policy decision (a no-brainer, really) to provide a user appearance preference for a static TOC, a pop-up TOC, or both.

Pyritohedral symmetry of the icosahedron
It is the unique polyhedral point group that is neither a rotation group nor a reflection group.

Borromean rings
https://www.mathunion.org/outreach/imu-logo/borromean-rings

https://archive.bridgesmathart.org/2008/bridges2008-63.pdf

The vertices of the regular icosahedron form five sets of three concentric, mutually orthogonal golden rectangles, whose edges form Borromean rings. In a Jessen's icosahedron of unit short radius one set of these three rectangles (the set in which the Jessen's icosahedron's long edges are the rectangles' long edges) measures $$2\times 4$$. These three rectangles are the shortest possible representation of the Borromean rings using only edges of the integer lattice.

Golden icosahedron
If we truncate eight vertices of a dodecahedron belonging to a cube we get a golden icosahedron (8 faces are equilateral triangles, the 12 others are golden triangles) which has tetrahedral symmetry (only three planes of symmetry). http://www.polyhedra-world.nc/tetra_di_.htm

https://robertlovespi.net/2022/01/09/the-pyritohedral-golden-icosahedron/

Reflections
The octahedron is unique among the Platonic solids in having an even number of faces (4 triangles) around each vertex. Consequently it is the only Platonic solid whose mirror planes run entirely through edges and do not divide any of its faces. Among the quasiregular solids, the cuboctahedron also has an even number of faces around each vertex (two triangles and two squares).

4-pyramid

Regular and quasiregular convex polyhedra
$\sqrt{2}$ ≈ 0.866

Characteristic tetrahedron
Because it is a quasiregular polyhedron, the cuboctahedron cannot be dissected into instances of a single characteristic tetrahedron that is an orthoscheme. As a quasiregular polyhedron, the cuboctahedron possesses all the square and equilateral triangle faces of the cube and octahedron for which it is named. It can be dissected into the component orthoschemes of that cube and octahedron; but these are two different orthoschemes. A left-handed orthoscheme and a right-handed orthoscheme of the same kind meet at each face: two mirror-image characteristic orthoschemes of the cube meet at each square face, and two mirror-image characteristic orthoschemes of the octahedron meet at each triangular face.

A characteristic tetrahedron must represent all the symmetries that exist in the polyhedron. It must be possible to construct the polyhedron by replicating its characteristic tetrahedron, so among other properties, the characteristic tetrahedron of the cuboctahedron would require edge lengths of $\sqrt{2}$, $\sqrt{1}$ , $\sqrt{2}$ , 1, $\sqrt{3}$, and $\sqrt{4}$ (as well as $\sqrt{5}$ unless the cuboctahedron could be dissected into characteristic tetrahedra with one vertex at its center). No such tetrahedron exists.

Cell rings
User:Cloudswrest/Regular polychoric rings





Chiral symmetry operations
{24/12}=12{2} $$^{-q7}$$ [16] 2𝝅 {2}

scratch
The same 720° isoclinic rotation takes each of its 1152 characteristic 5-cells to and through 11 other characteristic 5-cells, as an alternating sequence of left- and right-hand 4-orthoschemes (a sequence of 24 reflections), on a geodesic two-revolution orbit around the 3-sphere that covers 12 vertices (with each 5-cell occupying just one 24-cell vertex at a time).

The 12 stationary 4-orthoschemes visited by any one moving 4-orthoscheme in the course of a 720° isoclinic rotation are each cell-bonded to two others linearly, like the cars of a railroad train with alternating (4-dimensional) cars of two mirror-image shapes: the left-hand and right-hand forms of the same irregular 5-cell. The train runs on a circular geodesic track which it entirely fills, so it has no first or last car. The train of 5-cells forms a Möbius ring that wraps twice around the 24-cell without intersecting itself in any point. Although it visits half the 24-cell vertices just once, it consists of only 12 of its 1152 5-cells, and comprises only one 96th of its 4-dimensional content.

Our train analogy is not quite right, as train cars travel by translation not by reflection. Two reflections is a translation or rotation. A pair of adjacent 5-cells travels together by rotation. Perhaps each train car should be a left/right pair of 5-cells.

In the course of a 720° isoclinic rotation, the five vertices of each 5-cell occupy (in an alternating sequence of reflected 5-cells) the five vertex positions of 12 left-hand characteristic 5-cells and 12 right-hand characteristic 5-cells, while the orbiting 5-cell turns itself completely inside-out twice (the 5-cell itself rotating twice as it performs this orbit). Two revolutions (a 720° isoclinic rotation) does not quite take the moving 4-orthoscheme back to itself, however. It requires 96 such 720° isoclinic rotations (an orbit of 192 revolutions) to visit all 1152 4-orthoschemes and return the moving 4-orthoscheme to its original orientation.

If the 12-vertex rail running through the 24-cell vertices is a closed loop on a geodesic isocline as we have claimed, how many 5-cell-disjoint sequences of 12 5-cells run along that rail? It had better be 48, or the forgoing theory, at least, is falsified. But it is not inconceivable that 48 5-cells could surround each segment of the geodesic isocline. There are 48 5-cells meeting in each octahedral facet of the 24-cell, with their 3-orthoscheme bases (48 characteristic tetrahedra of the octahedron) packed around each octahedral cell center. How many 5-cells meet at each 24-cell vertex (also octahedron) vertex)? Perhaps also 48 (by duality)? Then there would be 48 directions for a train of reflecting 5-cells to depart from, or arrive at, each vertex.

Notice that each 720° ring of cell-bonded 4-orthoschemes has an exterior spine (the largest-radius rail of the 5-helix circular rail track) which is a sequence of the same 24 vertices of the 24-cell. The 48 ring sequences are 48 different orderings of those 24 vertices. Each isocline geodesic is like an entire fibration of the 24-cell, all by itself, a linearization of the 4-polytope into a single wrapped-around-twice fiber that visits all 24 vertices. It is not a Hopf fibration, which ic composed of multiple parallel great circle fibers. But as in any set of fibrations, each fibration runs in a different "direction" (related to the path of its particular isoclinic rotation) by virtue of having a different ordering of the same set of vertices. In the case of an "isoclinic fibration" there are no parallel fibers. What is the relationship between Hopf fibrations and "isoclinic fibrations"? We know that isoclines cross great circles, knitting them together, and we know that the relationship (in the 24-cell) is 4:1 (4 great circles of 6 hexagons versus a Mobius loop of 2 decagons). All fibrations are the same 24 vertices and differ only in their orderings and separation into non-intersecting parallel loops, but how exactly are these characterizing properties related in the Hopf fibration versus "isoclinic fibration" cases? Is each "isoclinic fibration" geometrically isomorphic to a Hopf fibration somehow, or is an "isoclinic fibration" a thing in its own right? Of course it is the former of these two possibilities which is correct, because the Clifford parallel Hopf fibers are related by an isoclinic rotation which brings them together. The Hopf fibers and the isoclines crossing between them are the warp and woof of the same fibration: that is their isomorphism. How many isoclines does each particular isoclinic rotation have in the 24-cell? Are they disjoint?

Why are we studying only isoclinic rotations re: orthoschemes? Because only in a double rotation do all the points except the 24-cell center move. Isoclinic rotation is the most symmetrical case; we study it first. But more generally, every translation or rotation in 4-space can be viewed as an isoclinic rotation by appropriate choice of reference frame; so every displacement in 4-space is an isoclinic rotation.

Heavy use of explanatory footnotes in the 600-cell article
Hi Beland. Thank you for reviewing the 600-cell article, and for your suggestion that the copious footnotes be simplified and improved, and perhaps pulled inline or moved to separate articles. The article does have a great many explanatory footnotes! The Notes section is 3/5 as large as all the rest of the article combined. That is indeed very unusual for a Wikipedia article, even a large article on a complex topic. So as the author of most of those notes, I feel it is incumbent upon me to try to explain why they are there. The heavy use of multiply-linked explanatory footnotes is certainly an outlier in the range of Wikipedia footnoting styles, but I hope to persuade you that in general they are a feature of this article, not a bug. I do agree that many of the concepts they explain deserve a separate Wikipedia article of their own, and I will be working on developing those articles in the future. It would be ideal to group the text of footnotes on the same topic into an article of its own, so the footnote references can be replaced with links.

The reason for the heavily interlinked footnotes is the special nature of the subject matter. This is an article about a 120-vertex object that lives in the fourth dimension, one of the most complex regular geometric objects in nature. It is bafflingly unfamiliar to nearly all human beings, because none of us has ever had the sensory experience of handling a four-dimensional object, and the only way we can visualize one is in our imagination. Even the illustrations in the article (which are excellent, by the way, and almost all the work of other Wikipedia editors than myself) are very hard to understand, because they are only illustrations of three-dimensional shadows and slices of the 600-cell, which is a four-dimensional object. The surface of the 600-cell is a curved three-dimensional space, the way the surface of the earth is a curved two-dimensional space, but even that is hard to illustrate or comprehend, because it is a curved non-Euclidean three-dimensional space, subtly different from the flat Euclidean three-dimensional every-day space we all live in. So to explain to the reader what he is looking at, even with these fine illustrations of parts of the thing, is quite a challenge.

If you read the 600-cell article section by section, line by line, I think you will find that you do not get far -- not through the first paragraph perhaps -- before you encounter a sentence that makes little sense to you, or a term or expression with which you are unfamiliar, even if you are a mathematician (and I am not). That is the reason for all the links and explanatory footnotes in the article. When you read a sentence and you don't quite get it, there is a footnote you can hover your mouse over, and a little post-it note pops up with more information, hopefully just what you need to have explained (if I have put the right footnote in the right place).

Where the concept or term has a Wikipedia article of its own, or a section of an article of its own, of course I use links in preference over footnotes, but there are hundreds of concepts and distinct mathematical terms required to enable a reader to visualize the fourth dimension, and not all of them have their own Wikipedia article (yet). Even where they do, an explanatory footnote may be needed in addition to the link, in order to explain the meaning of the term in this unique context; believe me, the 600-cell is really unique! Moreover, these ideas really can't be explained in isolation. It happens that the 600-cell is the archetype of many of those concepts, one of the few examples and in some cases the best example of a known object which has those properties. The only way I have been able to learn to visualize the fourth dimension at all is by studying the regular 4-polytopes as examples of four dimensional objects. So a quality article about the 600-cell is really an article explaining the fourth dimension, by way of the 600-cell as an example. An article which merely lists the 600-cell's properties and provides beautiful incomprehensible pictures of it would not be a high quality article. That's what we'd have if we deleted most of the footnotes, or drastically simplified them.

For the same reason that the 600-cell has a very complex geometry, the footnotes themselves have a very complex geometry. Those unfamiliar terms and strange concepts occur again and again in the article in distinct but related contexts. Whenever I write a note explaining a concept, I look for other places where the concept arises, such as in other notes about related concepts, and I add a reference to the footnote in those places, too. Just as an article may have many links to it, an explanatory note is a mini-article which may have many references to it. I hope that I have linked these notes to each other in the proper way, so that the relationships among the concepts and geometric objects is mirrored in the topology of the notes themselves, and the notes can explain the relationships among themselves to the reader. In this case, the body of footnotes itself is a polytope. Not a regular polytope to be sure, and probably of more than four dimensions in some places, but a complex geometic object to be explored. It is entirely up to the reader how to do that exploring. It is up to him whether he clicks on the footnote-within-a-footnote, just as it is up to him whether he clicks on a footnote in the first place. But if he needs to, he can explore the concept the footnote is about in depth, reviewing several notes-within-notes before he satisfies himself that he understands the sentence of the article he was reading that stumped him.

So I invite you to do more reviewing of the article and its footnotes, in greater depth if you have the interest and the time, and to tell me which footnotes were helpful to you, which were confusing or unhelpful, and which were unnecessary or redundant or badly linked. I want to improve them, and that always means simplifying them, making them more concise and briefer, wherever I can find a way to do so. You can help me to do that. Tell me if the fourth dimension makes sense to you, with or without the footnotes. Tell me where you need another footnote to explain something that is not yet explained! Dc.samizdat (talk) 09:56, 21 March 2024 (UTC)

Chiral symmetry operations
A symmetry operation is a rotation or reflection which leaves the object in the same orientation, indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two reflections, in a distinct pair of non-parallel mirror planes.

Pictured are sets of disjoint great circle polygons, each in a distinct central plane of the 600-cell. For example, 4{30/3}=12{10} is an orthogonal projection of the 600-cell picturing 3 of its 72 great decagon planes, each of which can be seen as 4 disjoint great decagons. The 12 great decagons lie Clifford parallel to the projection plane and to each other, and collectively constitute a discrete Hopf fibration of 12 non-intersecting great circles which visit all 120 vertices just once.

Each row of the table describes a class of distinct rotations. Each rotation class takes the left planes pictured to the corresponding right planes pictured. The vertices of the moving planes move in parallel along the polygonal isocline paths pictured. For example, the $$[32]R_{q7,q8}$$ rotation class consists of [32] distinct rotational displacements by an arc-distance of $\sqrt{6}$ = 120° between 16 great hexagon planes represented by quaternion group $$q7$$ and a corresponding set of 16 great hexagon planes represented by quaternion group $$q8$$. One of the [32] distinct rotations of this class moves the representative vertex coordinate $$(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})$$ to the vertex coordinate $$(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})$$.

For example, the $$[144]R_{q14,-q14}$$ rotation class consists of [144] distinct rotational displacements by an arc-distance of $𝝅⁄2$ = 36° between 72 great decagon planes represented by quaternion group $$q14$$ and a corresponding set of 72 great decagon planes represented by quaternion group $$-q14$$. One of the [144] distinct rotations of this class moves the representative vertex coordinate $$(\tfrac{\phi}{2},\tfrac{1}{2},\tfrac{\phi^{-1}}{2},0)$$ to the vertex coordinate $$(-\tfrac{\phi}{2},-\tfrac{1}{2},-\tfrac{\phi^{-1}}{2},0)$$.

In a rotation class $$[d]{R_{ql,qr}}$$ each quaternion group $$\pm{q_n}$$ may be representative not only of its own fibration of Clifford parallel planes but also of the other congruent fibrations. For example, rotation class $$[4]R_{q7,q8}$$ takes the 4 hexagon planes of $$q7$$ to the 4 hexagon planes of $$q8$$ which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind, all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes $$[4]R_{-q7,-q8}$$, $$[4]R_{q8,q7}$$ and $$[4]R_{-q8,-q7}$$. The name $$[16]R_{q7,q8}$$ is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of $$[32]R_{q7,q8}$$ which has [32] distinct rotational displacements rather than [16] because there are two chiral ways to perform any class of rotations, designated its left rotations and its right rotations. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes. Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously, through a characteristic rotation angle. For example, the $$[32]R_{q7,q8}$$ rotation moves all [16] hexagonal planes at once by $𝝅⁄3$ = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same orientation, passing through all 4 planes of the $$q7$$ left set and all 4 planes of the $$q8$$ right set once each. The picture in the isocline column represents this union of the left and right plane sets. In the $$[32]R_{q7,q8}$$ example it can be seen as a set of 4 Clifford parallel skew hexagrams, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.

Bilbo and Frodo's birthday
Tolkien The Lord of the Rings  Book I, Chapter 1

Twelve more years passed.... Bilbo was going to be eleventy-one, 111, a rather curious number ... and Frodo was going to be thirty-three, 33, the date of his ‛coming of age’.

Circumference of isoclines
The circumference of a 30-cell ring varies from 10 edge lengths on a decagon great circle, to 30 edge lengths along its helical Petrie polygon. The circumference of an isocline along its chords (which are 24-cell edges) is 15. The shortest path around a 30-cell ring on 600-cell edges is 20 edge-lengths

Each vertex of the 30-cell ring is 20 edge lengths distant from itself around the isocline by the shortest path along 600-cell edges, which is a strip of triangles edge-bonded at those 20 edges; but it is 21 tetrahedra distant from itself by the shortest chain of edge-bonded tetrahedra. Perhaps this difference of 1 edge length occurs for the same reason that Phineas Fogg passed through 81 distinct days (sunrise-sunset pairs) on his journey Around the World in 80 Days. Phineas Fogg performed one slow forward somersault in the course of his 80 day journey, in addition to the 80 times he was rotated all the way around the planet. Each day he hastened to his sunset a little faster than the planet would normally have brought him to it, and each of his days was a little shorter than 24 hours. He rotated through 81 distinct "solar" rotations (which of course are not really rotations of the sun, but rotations of ourselves about the earth with respect to the sun). We can see his extra day as a consequence of his slow extra somersault, as he himself was rotated in place once in the course of his 80-rotation journey. In a single 360° isoclinic rotation of the 600-cell, each of its 600 tetrahedral "passengers" is itself rotated in place once, in addition to being rotated all the way around the 600-cell. Each tetrahedron is rotated in the first place at each of the 10 36°x36° steps of the rotation to another tetrahedron two edge-lengths away, in a different rotational orientation.

for triacontagon note
Of course there is also a unit-radius great circle in this central plane (completely orthogonal to a decagon central plane), but as a great circle polygon it is a 0-gon, not a 30-gon, because it intersects none of the points of the 600-cell.

Isoclines: g and h
From Coxeter element:


 * There are relations between the order g of the Coxeter group and the Coxeter number h:
 * * [p]: 2h/gp = 1
 * * [p,q]: 8/gp,q = 2/p + 2/q -1
 * * [p,q,r]: 64h/gp,q,r = 12 - p - 2q - r + 4/p + 4/r
 * * [p,q,r,s]: 16/gp,q,r,s = 8/gp,q,r + 8/gq,r,s + 2/(ps) - 1/p - 1/q - 1/r - 1/s +1
 * For example, [3,3,5] has h=30, so 64*30/g = 12 - 3 - 6 - 5 + 4/3 + 4/5 = 2/15, so g = 1920*15/2 = 960*15 = 14400.
 * For example, [3,3,5] has h=30, so 64*30/g = 12 - 3 - 6 - 5 + 4/3 + 4/5 = 2/15, so g = 1920*15/2 = 960*15 = 14400.
 * For example, [3,3,5] has h=30, so 64*30/g = 12 - 3 - 6 - 5 + 4/3 + 4/5 = 2/15, so g = 1920*15/2 = 960*15 = 14400.

Each isocline is a {30/2}=2{15} triacontagram2. They come in chiral pairs. Are there 3840, 1920, 960 or 480 of them?

Falsified theory
Any five 24-cells which can be reached by either 5-click simple rotation by itself have disjoint vertices.

We can give the 25 24-cells physical addresses of the form (i, j) corresponding to their inclination from an origin 24-cell, where i and j are integers between 0 and 4 corresponding to multiples of $𝝅⁄4$ in the two orthogonal planes. The 25 overlapping 24-cells are thus arranged logically by address (not physically) in a 5 x 5 array. The five 24-cells in each row of this array (any five 24-cells with the same i) are disjoint: they have disjoint sets of vertices that together account for all 120 vertices of the 600-cell. The five 24-cells in each column of the array (any five with the same j) are similarly disjoint. No other set of five 24-cells is disjoint; these are the only ten ways to partition the 25 24-cells into five disjoint 24-cells.

Pentimento
Notice that the 600-cell has two pentagons inscribed in each decagon (as the 24-cell has two triangles inscribed in each hexagon). The pentagon's $𝝅⁄3$ edge chord falls between the $𝝅⁄3$ hexagon edge chord and the $𝝅⁄2$ square edge chord in length. The 600-cell has added a new interior boundary envelope (of cells made of pentagon edges), which has a short radius between those of the 24-cells' envelopes of octahedra (made of $𝝅⁄3$ hexagon edges) and the 16-cells' envelopes of tetrahedra (made of $𝝅⁄4$ square edges). Consider also the $𝝅⁄2$ = φ and $𝝅⁄4$ chords. These too will have their own characteristic face planes and interior cells, and their own envelopes, of some kind not found in the 24-cell. The 600-cell is not merely a new skin of 600 tetrahedra over the 24-cell; it also inserts new features deep in the interstices of the 24-cell's interior structure, which it inherits in full, compounds five-fold, and then elaborates on.

Euler's identity
Euler's identity $$e^{i\pi}=-1$$ has a geometric representation as a rotation in the complex plane. Euler's identity can be interpreted as saying that rotating any point $$\pi$$ radians around the origin has the same effect as reflecting the point across the origin. Similarly, the derivative equation $$e^{2\pi i} = 1$$ can be interpreted as saying that rotating any point by one turn around the origin (in 2-dimensional space) returns it to its original position.

Euler's identity captures the relation between the Coxeter group operations rotation and reflection, in Euclidean spaces of any number of dimensions.

Rotations in 4-dimensional Euclidean space are rotations in the quaternion Cartesian space, where the equation $$e^{4\pi{i}}=1$$ can be interpreted as an analogous assertion.

...

Geodesic rectangles


.. great rectangles in .. ☐ planes



.. great rectangles in .. ☐ planes

Characteristic orthoscheme
Like all regular convex polytopes, the dodecahedron can be dissected into an integral number of disjoint orthoschemes, all of the same shape characteristic of the polytope. A polytope's characteristic orthoscheme is a fundamental property because the polytope is generated by reflections in the facets of its orthoscheme. The orthoscheme occurs in two chiral forms which are mirror images of each other. The characteristic orthoscheme of a regular polyhedron is a quadrirectangular irregular tetrahedron.

The faces of the dodecahedron's characteristic tetrahedron lie in the dodecahedron's mirror planes of symmetry. The dodecahedron's symmetry group is denoted Ih or H3. The dodecahedron and its dual polytope, the icosahedron, have the same symmetry group but different characteristic tetrahedra.

The characteristic tetrahedron of the regular dodecahedron can be found by a canonical dissection of the regular dodecahedron which subdivides it into 120 of these characteristic orthoschemes  surrounding the dodecahedron's center. Five left-handed orthoschemes and five right-handed orthoschemes meet in each of the dodecahedron's twelve faces, the ten orthoschemes collectively forming a pentagonal pyramid with the dodecahedron's pentagonal face as its equilateral base, and its apex at the center of the dodecahedron.

If the dodecahedron has radius 𝒍 = 2, its characteristic tetrahedron's six edges have lengths $$\sqrt{\tfrac{4}{3}}$$, $$\tfrac{1}{\phi}$$ , $$\sqrt{\tfrac{1}{3}}$$ around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁), plus $$\sqrt{2}$$ , $$1$$ , $$\sqrt{\tfrac{2}{3}}$$ (edges that are the characteristic radii of the octahedron). The 3-edge path along orthogonal edges of the orthoscheme is $$1$$, $$\sqrt{\tfrac{1}{3}}$$ , $$\sqrt{\tfrac{2}{3}}$$ , first from an octahedron vertex to an octahedron edge center, then turning 90° to an octahedron face center, then turning 90° to the octahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a 90-60-30 triangle which is one-sixth of an octahedron face. The three faces interior to the octahedron are: a 45-90-45 triangle with edges $$1$$, $$\sqrt{2}$$ , $$1$$ , a right triangle with edges $$\sqrt{\tfrac{1}{3}}$$ , $$1$$ , $$\sqrt{\tfrac{2}{3}}$$ , and a right triangle with edges $$\sqrt{\tfrac{4}{3}}$$ , $$\sqrt{2}$$ , $$\sqrt{\tfrac{2}{3}}$$.

Characteristic orthoscheme
Every regular 4-polytope has its characteristic 4-orthoscheme, an irregular 5-cell. The characteristic 5-cell of the regular 600-cell is represented by the Coxeter-Dynkin diagram, which can be read as a list of the dihedral angles between its mirror facets. It is an irregular tetrahedral pyramid based on the characteristic tetrahedron of the regular tetrahedron. The regular 600-cell is subdivided by its symmetry hyperplanes into 14400 instances of its characteristic 5-cell that all meet at its center.

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 600-cell). If the regular 600-cell has unit radius and edge length $$\text{𝒍} = \tfrac{1}{\phi} \approx 0.618$$, its characteristic 5-cell's ten edges have lengths $$\sqrt{\tfrac{2}{3\phi^2}}$$ , $$\sqrt{\tfrac{1}{2\phi^2}}$$ , $$\sqrt{\tfrac{1}{6\phi^2}}$$ around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁), plus $$\sqrt{\tfrac{3}{4\phi^2}}$$ , $$\sqrt{\tfrac{1}{4\phi^2}}$$ , $$\sqrt{\tfrac{1}{12\phi^2}}$$ (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the characteristic radii of the regular tetrahedron), plus $$1$$ , $$\sqrt{\tfrac{5 + \sqrt{5}}{8}}$$ , $$\sqrt{\tfrac{\phi^2}{3}}$$ , $$\sqrt{\tfrac{\phi^4}{8}}$$ (edges which are the characteristic radii of the 600-cell). The 4-edge path along orthogonal edges of the orthoscheme is $$\sqrt{\tfrac{1}{2\phi^2}}$$, $$\sqrt{\tfrac{1}{6\phi^2}}$$ , $$\sqrt{\tfrac{1}{4\phi^2}}$$ , $$\sqrt{\tfrac{\phi^4}{8}}$$ , first from a 600-cell vertex to a 600-cell edge center, then turning 90° to a 600-cell face center, then turning 90° to a 600-cell tetrahedral cell center, then turning 90° to the 600-cell center.

Coordinates
({±$𝝅⁄3$, ±$𝝅⁄2$, ±$𝝅⁄4$, ±$𝝅⁄3$}) ([±$𝝅⁄2$, ±$𝝅⁄3$, ±$𝝅⁄5$, ±$𝝅⁄2$]) ({±$𝝅⁄5$, ±$𝝅⁄3$, ±$𝝅⁄2$, ±$𝝅⁄5$}) ([0, ±$𝝅⁄3$, ±$𝝅⁄2$, ±$𝝅⁄2$]) ([±$𝝅⁄3$, ±$𝝅⁄2$, ±$𝝅⁄3$, ±$\sqrt{2}$])

Units
planck (energy/wavelength) constant: $$h = 1$$

(reduced) planck (action or angular momentum) constant: \hbar = h/{2\pi} = 1 \approx $$6.626 \times 10^{-34}$$ joule/hertz (energy/cycle)

speed of light in $$R_3$$: $$c = 1 = $$

planck time: ~10^-44 seconds

planck length: ~10^-35 meters

planck mass (gravitational constant): $$G = 1 = $$

planck temperature

symmetry operation rate:

H2O covalent bond @ 104.5°: ~100 pm = .1 nm = 1 x 10-10 meters

compton radius of electron: 3.86 x 10-11 cm = 10-13 meters

Research articles
Articles I write or contribute to which contain some original research, and so cannot be published as Wikipedia articles, are hosted at Wikiversity instead.

See also my Wikiversity User page.

Koca
Koca's slides from a Bangalore conference summarizing his studies of uniform 4-polyopes; he has described quite a few hitherto-nondescript 4-polytopes, such as the 720-point 720-cell (600 octahedra and 120 isosahedra), and many nondescript duals of known uniform 4-polytopes; some of these polytopes have vertices on two different concentric 3-spheres, so they reveal relationships between 4-polytopes of different radii https://slideplayer.com/slide/8639634/

Daniel Piker
https://spacesymmetrystructure.wordpress.com/2008/12/11/4-dimensional-rotations/#more-160

https://spacesymmetrystructure.wordpress.com/links/4-dimensional-rotations-page2/

https://spacesymmetrystructure.wordpress.com/links/4-dimensional-rotations-page3/

https://spacesymmetrystructure.wordpress.com/links/4-dimensional-rotations-page4/

As the invariant axis of a rotating 2-sphere is dimensionally analagous to the invariant plane of a rotating 3-sphere, the poles of the rotating 2-sphere are dimensionally analogous to linked great circles on the 3-sphere. By dimensional analogy, each 1D point in 3D lifts to a 2D line in 4D, in this case a circle. The two antipodal rotation poles lift to a pair of circular Hopf fibers which are not merely Clifford parallel and interlinked, but also completely orthogonal. The invariant great circles of the 4D rotation are its poles. In the case of an isoclinic rotation, there is not merely one such pair of 2D poles (completely orthogonal Hopf great circle fibers), there are many such pairs: a finite number of circle-pairs if the 3-sphere fibration is discrete (e.g. a regular polytope with a finite number of vertices), or else an infinite number of orthogonal circle-pairs, entirely filling the 3-sphere. Every point in the curved 3-space of the 3-sphere lies on one such circle (never on two, since the completely orthogonal circles, like all the Hopf great circle fibers, do not intersect). Where a 2D rotation has one pole, and a 3D rotation of a 2-sphere has 2 poles, an isoclinic 4D rotation of the 3-sphere has nothing but poles, an infinite number of them. In a discrete 4-polytope, all the Clifford parallel invariant great polygons of the rotation are poles. In one full revolution of such a rotation, every point in the space loops exactly once through one of these pole-circles. The circles are arranged with a surprising symmetry, so that each pole-circle links with every other pole-circle, like an infinitely dense 4D fabric of chain mail, and no 2 circles ever intersect.

Paulo Freire's polytope models
https://www3.mpifr-bonn.mpg.de/staff/pfreire/polyhedra/

Includes models of all the various compounds of the regular polychora. The 120-cell has a lot of them.

https://www3.mpifr-bonn.mpg.de/staff/pfreire/polyhedra/compounds.htm