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Comprehending the shapes of time through uniform polychora
Enhancing Strategic Discourse Systematically using Climate Metaphors (Part #6)
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Analogy and symmetry: In the purely spatial terms of geometry, a 4-polytope (also called a polychoron, polycell, or polyhedroid) is a 4-dimensional polytope. It is a connected and closed figure, as with its 2-dimensional analogues: the polygon, and the 3-dimensional the polyhedron. The emphasis is here is on the association of the 4th dimension with time -- a preoccupation seldom (if ever) considered in the literature on 4-polytopes, although significant to consideration of spacetime (Yasha Neiman, Causal cells: spacetime polytopes with null hyperfaces, 2012). Whether as a spatial or temporal dimension, its comprehension and representation poses particular challenges -- since such forms cannot be "seen" in 3-dimensional space, as is evident, by analogy, in the difficulty of representing 3-dimensionality on a flat surface.
Just as symmetry makes it easier to understand and distinguish different 3D polyhedra, some degree of comprehension is however enabled by the distinctions between the uniform polychora, as variously documented (George Olshevsky, Uniform Polytopes in Four Dimensions, 1997-2006; Jonathan Bowers, Uniform Polychora, 2006; Jonathan Bowers, Uniform Polychora and Other Four Dimensional Shapes, 2014). An extensive classification of uniform 4-polytopes is also presented by Wikipedia, also variously depicted. Note also the Wikia List of Polychora by Type. Valuable clarifications are provided by Richard Klitzing (Polytopes and their Incidence Matrices, 2015)
Enumeration: Of direct relevance to the argument here, 64 convex uniform 4-polytopes are recognized, and presented by Wikipedia according to the classification of George Olshevsky. These include 6 regular convex 4-polytopes (excluding the infinite sets of the duoprisms and the antiprismatic hyperprisms). Somewhat ironically, this suggests that these might well be usefully represented as vertices on some of the 3D polyhedra depicted above.
Of further relevance to the discussion below, Bowers clusters 1845 uniform polychora (including the non-convex) into 29 categories. This set is generally thought to be complete, but not proven to be so. Given the limited familiarity with these forms, it is appropriate to stress the controversy surrounding use of terms such as polychoron, the various naming conventions proposed, and the variety of forms distinguished to date [see Wikipedia discussion].
The Stella4D variant of the Stella Polyhedron Navigator (used for the depictions above) generates remarkable interactive views of known uniform polychora including the 64 convex forms and the infinite prismatic families [see Stella 4D Manual, with its relevant discussion of the Fourth Dimension]. The screen shots below were made with this facility.
Terminology: A flavour of these strange forms -- of relevance to any discussion of a time-bound universe and the "shapes of time" -- is offered by the 29 categories distinguished by Bowers and featuring in the Stella 4D polytope library. Fundamental to the controversy are the deprecated (but memorable) neologisms promoted by Bowers and the (essentially unmememorable) conventions of mathematics helpfully indicated by George Olshevsky (Glossary for Hyperspace, 2001), clarifying the use of acronyms promoted by Bowers (Naming the convex uniform polychora). Note the formal complete list of 64 non-prismatic convex uniform polychora in Wikipedia. Variants of the terminology are also reconciled by Richard Klitzing (Regular Polytopes)
The extraordinary terminology might be considered especially relevant to imaginative engagement with such forms. A language for the shapes of time to which governance is obliged to respond? Or is it to be assumed that administrative English is adequate to the governance challenges of navigating through time?
As with multiple distinctions for a phenomenon in some languages, do the shapes of time merit similar distinctions?
As indicated by its extensive use in the checklists above and below, a vertex figure (abridged to "verf"), or vertex configuration, needs to be understood as the shape obtained from drawing a small hypersphere around any vertex of a uniform polytope (What Does a 4-Dimensional Sphere Look Like? 2002). Loosely speaking, it is the (n-1)-dimensional polytope formed at the stump when a corner is truncated from an n-dimensional polytope or compound polytope -- namely the cross-section of an n-dimensional polytope or compound polytope very close to a vertex. The vertex figure shows exactly how a polytope's facets fit together to surround a vertex.
Categories of uniform polychora (notably including the non-convex)
| 29 Categories of uniform polychora (notably including the non-convex) (reproduced from Jonathan Bowers, Uniform Polychora, 2006, with minor editing and addition of emphases) | |
Category 1: Regulars (Polychora 1-17): These are the 16 regular polychora plus the only faceting of hex - "tho" - there are 17 polychora here. Verfs are regular polyhedra, and in tho's case the verf is a thah. [images} Category 2: Truncates (Polychora 18-38): These are the truncated and quasitruncated polychora, there are also three ditrigonary truncates. Verfs are pyramids of regular polygons or semiregular polygons. [images] Category 3: Triangular Rectates (Polychora 39-59): These are the rectified pen, tes, ico, hi, sishi, gaghi, and gogishi and their two primary facetings. There are 7 regiments represented here with three members each (rit and rico has more regiment members mentioned in Cat. 12 and Cat. 6 respectively). There verfs are triangle prisms along with their facetings. [images] Category 4: Ico Regiment (Polychora 60-72): These are the facetings of ico, one of them, ihi, has pyrito-ico symmetry, 6 have tessic symmetry, while the other 6 have demitessic symmetry. Verfs are facetings of the cube. There's also a prominent compound called "Gico".[images] Category 5: Pentagonal Rectates (Polychora 73-132): These are the polychora that belong to the rox army composed of the rox, righi, ragishi, and rigfix regiments. The 5 regiments earch have 15 members, there are also two coinciding members and five exotic members in each regiment, which are no longer counted as polychora. The verfs are variant facetings of variant pentagon prisms. [images] Category 6: Sphenoverts (Polychora 133-297): These are the cantellates (also called small rhombates): of the polychora along with others with similar verfs. Verfs are wedges and their facetings, each of the 24 regiments have 7 members(rico has had 3 members already counted in cat. 3). Sirgax belongs to this group. Category 7: Bitruncates (Polychora 298-306): These nine polychora (deca, tah, cont, xhi, shihi, dahi, gixhi, gic, and ghihi): are the bitruncates, they all have disphenoid verfs. Cont, gic, and deca have only one type of cell. There are also two fissary cases sitphi and gitphi which have only one type of cell, their verfs are compounds of three disphenoids. [images] Category 8: Grombates (Polychora 307-329): These 23 polychora are also known as the great rhombates and their kin. There verfs are scalenoids (a scalene like disphenoid). Category 9: Omnitruncates (Polychora 330-351): These 22 polychora are also known as the maximized polychora. Their verfs are irregular tets. Category 10: Prismatorhombates (Polychora 352-441): These 90 polychora are grouped into 30 regiments of 3, they seem to be quite attractive. Their verfs are trapyrs and facetings. One of my favorites is giphihix. Category 11:Antipodiumverts (Polychora 442-481): These 40 polychora are grouped into 5 regiments of 7 and 1 regiment of 5. They have triangle antipodium shaped verfs along with facetings. The small prismates, like sidpith, belong here. There are some scaliforms in the sidpith regiment also. [images] Category 12: Podiumverts (Polychora 482-511): These 30 polychora are grouped into 4 regiments of 7 and the extra two members of the rit regiment (sto and gotto). Their verfs are triangle podiums and their facetings, sixhidy belongs here. Previously known as frustrumverts. There are some scaliforms amongst the gittith regiment. [images] Category 13: Spic and Giddic Regiments (Polychora 512-551): These 40 polychora are split into 2 regiments of 20. Spic has 48 octs and 96 trips as cells, Giddic has 48 octs and 48 quiths as cells. They both have a sort of square antiprism verf. Each regiment also has 2 fissary members. Category 14: Skewverts (Polychora 552-611): These 60 polychora are split into 4 regiments of 15, their verfs are skewed wedges and facetings. Many of these are very intricate. The regiments are skiviphado (tessic), gik vixathi, sik vipathi, and skiv datapixady (last three are hyic). Category 15: Afdec Regiment (Polychora 612-664): The afdec regiment has 53 members plus one fissary member called affic which has 48 cotcoes for cells. Afdec has 48 coes and 48 goccoes for cells, its verf is rectangle trapezoprism (which I first called an antifrustrum). Category 16: Affixthi Regiment (Polychora 665-763): The affixthi regiment has 99 members plus one fissary member (affidhi). Affixthi's cells are 600 octs, 120 dids, 120 gidditdids, and 120 gaddids. Its verf is similar to afdec's except that the bases have different shaped rectangles (an oct verf and a did verf). | Category 17: Sishi Regiment (Polychora 764-777): Sishi is the regular small stellated 120-cell which has a dodecahedron shaped verf, these 14 polychora are its non-regular, non-swirlprism facetings. There are also 2 fissaries and several exotic-celled members. Three of these have verfs shaped like the three ditrigonary polyhedra. Paphicki and paphacki (the small and great prismasauri) are also here. [images] Category 18: Ditetrahedrals (Polychora 778-888): These polychora all have 600 vertices, there are 3 regiments of 37, each regiment also has 4 fissaries, 20 exotic-celled cases, and 11 coincidic cases. The three regiments are the sidtaxhi, dattady, and gadtaxady regiments. Sidtaxhi's cells are 600 tets and 120 sidtids, verf is tut like. Dattady's cells are 120 gissids and 120 sidtids, verf is also tut like. Gadtaxady's cells are 120 gissids, 600 tets, and 120 gids, verf is a golden cuboctahedron (looks like a co, but squares are turned to golden rectangles). Sitphi and Gitphi can also go here as well as a similar compound which shows up in the dattady regiment. Category 19: Prisms (Polychora 889-962): These 74 polychora are the prisms of 74 of the 75 uniform polyhedra (we excluded the cube since the cube prism is the tesseract). Verfs are pyramids of the polyhedron verfs. Category 20: Miscellaneous (Polychora 963-984, 1846-1849): These 22 polychora include iquipadah, gaquipadah, the newly discovered ondip type, the antiprisms, snubs, and swirlprisms. The grand antiprism (gap) belongs here. This set contains all sorts of odd shaped polychora. Several scaliforms would fit amongst these, since many are swirlprisms.[images] Category 21: Padohi Regiment (Polychora 985-1065): The padohi regiment now has 81 members (it once had 354, where most of them were exotic-celled or coincidic). If we added the fissaries back in, the padohi regiment would double in size. Padohi's verf is a pentagonal antipodium. It's cells are 120 sissids, 120 ikes, 720 stips, and 1200 trips. Category 22: Gidipthi Regiment (Polychora 1066-1146): The gidipthi regiment also has 81 members since it is the conjugate of the padohi regiment. It's verf is a pentagonal podium. It's cells are 120 sissids, 120 ikes, and 120 gaddids. Many of its members are very intricate. Category 23: Rissidtixhi Regiment (Polychora 1147-1303): The rissidtixhi regiment (sometimes called the rissids) has 157 members (once it had 316) it also has a few fissary cases. It's verf is a ditrigon prism. Cells are 120 sidtids, 600 octs, and 120 gids. Some strange looking verfs show up in this regiment. Category 24: Stut Phiddix Regiment (Polychora 1304-1382): The stut phiddix regiment now has 79 members (once it had 238). Its verf is a triangle cupola, cells are 600 tets, 120 sidtids, 600 coes, and 720 stips. There are some beautiful polychora amongst the stuts. Category 25: Getit Xethi Regiment (Polychora 1383-1461): The getit xethi regiment also has 79 members (once it had 238). It's verf is a triangle cupola, cells are 600 tets, 120 sidtids, 120 gaddids, and 120 quit gissids. Category 26: Blends- (Polychora 1462-1473): These 12 polychora belong to the strange sabbadipady regiment which also contains 4 fissaries, its cells are 120 gissids, 720 stips, 720 pips, and 120 quit sissids. The verf looks like a triangle antipodium with a pyramid stuck on it's base. Some of the facetings have some really odd verfs. Category 27: Sidtaps and Gidtaps (Polychora 1474-1491): These 18 polychora are split into 2 regiments of 9, there were also some exotics here two as well as scaliforms. The sidtaps (or the sadsadox regiment) are based off of the blended compound of 10 roxes (which is no longer a compound, but a true polychoron). Likewise the gidtaps (gadsadox regiment) is based off of the blended compound of 10 raggixes. These are also known as the baby monster snubs and are related to the idcossids and dircospids. The verfs are facetings of a 2-pip blend Category 28: Idcossids (Polychora 1492-1668): The idcossids once had 2749 polychora, but nearly all of them were exotic-celled or coincidic, etc., now only 177 are left (however there are scaliforms here). Even the polychoron that the idcossids were named after was exotic-celled. The idcossids are based off of the 10-padohi compound, were the verfs are facetings of a pentagonal antipodium duo-combo. Many of these have millions of pieces. I now consider sadros daskydox as the head of this regiment (the conjugate of gadros daskydox). Category 29: Dircospids (Polychora 1669-1845): The dircospids are based off of the 10-gidipthi compound, only 177 are left as true polychora plus many scaliforms. The verfs are facetings of a pentagonal podium duo-combo. Gadros daskydox is considered the head. The dircospids are so far the most complex of the uniform polychora. |
64 convex uniform polychora: Oshevsky helpfully distinguishes these as follows, notably with indications of the (many) alternative terminologies employed, including the acronyms of Bowers (as employed in the Stella 4D library of polytopes). The need for such acronyms is evident from the unmemorability of the formal names, however meaningful they may be to some. Ironically the acronyms recall the succinct metaphorical terms (in Chinese) for each of the 64 hexagram-encoded conditions of the Book of Changes.
| Complete list of 64 non-prismatic convex uniform polychora (each item preceded, for convenience, by addition of the mnemonic acronym of Jonathan Bowers with an indication of the corresponding Category appended, if identifiable from Stella 4D) ) | |
Section 1: Polychora #1-9: based on the pentachoron (5-cell) 1: Pen (Pentachoron) Section 2: Polychora #10-21: based on the tesseract (hypercube) and hexadecachoron (16-cell) 10: Tes (Tesseract) Section 3: Polychora #22-31: based on the icositetrachoron (24-cell): 22: Ico (Icositetrachoron) [Cat. 1]
| Section 4: Polychora #32-46: based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) 32: Hi (Hecatonicosachoron) [Cat. 1] Section 5: Polychoron #47: anomalous non-Wythoffian polychoron 47: Gap (Grand antiprism) Section 6: Polychora #48-64: prismatic polychora: and infinite sets 48: Tepe (Tetrahedral prism) [Cat. 19] Section 7: derived from glomeric tetrahedron B4: all duplicates of prior polychora |
Polychora with 64 vertices: As an exercise in understanding how 64 conditions of changing climate might be coherently understood, the polychora with 64 vertices are presented below in separate slide shows. Appropriate to the inherent complexity of a condition of potential change, the vertices in 4D should be understood as having a dynamic temporal component, despite their conventional depiction as seemingly static points in the Category 12 images (on the left below). This is emphasized by the depiction of the duals of the Category 11 set (on the right below). In this case the 64 vertices take the form of cells, a display choice made because the non-dual form is visually of relatively little visual interrest.
| Potential polychora mapping surfaces for 64 conditions | |
| Category 12: Gittith regiment plus (edges added in some images) | Category 11: Sidpith duals plus (details are for non-dual variant) |
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Mapping surfaces with 192 or 384 vertices or edges: In an effort to obtain a means of comprehending the set of 384 transformations indicated by the pattern of the Book of Changes, the following slide shows indicate some of the forms capable of holding this pattern of 6x64 or 3x64 as a mapping. These images are especially relevant to further investigation of possibilities indicated by the drilled truncated cube (mentioned above, and discussed further below).
| Mapping surfaces for 384 vertices / edges or 192 vertices / edges | |
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