CRF polychora discovery project (Meta, 13)

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<[#ontology [kind meta] [cats 4D CRF Polytope]]>
{{selfref|See also: [[List of convex regular-faced polyhedra]]}}
{{selfref|See also: [[List of convex regular-faced polyhedra]]}}
This page documents an ongoing project to discover as many [[Polytope#Convex regular-faced polytope|CRF polychora]] as possible, and perhaps as a long-term goal prove that every CRF polychoron has been found.
This page documents an ongoing project to discover as many [[Polytope#Convex regular-faced polytope|CRF polychora]] as possible, and perhaps as a long-term goal prove that every CRF polychoron has been found.
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== Summary ==
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Other than infinite series, the non-Johnson CRF polyhedra are the regular polyhedra and the Archimedean polyhedra. The Johnson solids can be roughly divided into four categories: prismatoids (includes pyramids, cupolae, rotundae, and elongated/bi/gyro forms of the above), augmented polyhedra, diminished polyhedra and [[crown jewel]]s. We will use the same categorizations here.
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{| class='wikitable'
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!Class||Expression||Value||Total
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|-
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|Convex uniform polychora||9 + 9 + 12 + 15 + 17 + 2||64||64
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|-
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|Prisms of Johnson solids||92||92||156
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|-
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|Prismatoid forms||13×4 - 2||50||206
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|-
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|Cupolae of regular polyhedra||3×7 + 2×4 - 1 + 6||34||240
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|-
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|Bicupolic rings||3×3||9||249
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|-
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|Ursachora||2×3||6||255
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|-
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|Convex non-uniform scaliform||4||4||259
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|-
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|Rotundae||5×4||20||279
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|-
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|Duoprisms augmented with pyramids||See below||1633||1912
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|-
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|Duoprisms augmented with n-gonal magnabicupolic rings||See below||>1633||>3545
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|}
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== Convex uniform polychora ==
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Richard Klitzing enumerated a list of 177 [[segmentochora]], which are the orbiform CRF polychora. However, these overlap with several different categories below, so the segmentochora will not be considered a category of their own. <span class='bad'>TODO: find out how many segmentochora are not yet counted below.</span>
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The first 64 CRF polychora are the convex uniform polychora, which can be divided up into:
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*9 pyromorphs,
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*9 xylomorphs,
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*12 stauromorphs (not 15, because three were already covered as xylomorphs),
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*15 rhodomorphs,
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*17 prisms of convex uniform polyhedra (not 18, because one was already covered as the tesseract, a stauromorph),
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*the [[snub demitesseract]] and the [[grand antiprism]].
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== Monostratic polychora ==
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==Discovery index (D numbers)==
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Monostratic polychora are those whose vertices lie in two parallel hyperplanes. These are the simplest cases of CRF polychora to study, and they are also useful for augmenting larger polychora to produce more CRFs.
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=== Prisms of Johnson solids ===
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As of February 2014, new CRF discoveries are assigned a [[discovery index]] (aka ''D number''), as a way of uniquely identifying the discovery without committing to a specific categorization or naming of it, which may not be feasible due to insufficient information at the time of discovery.
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There are 92 [[Johnson solids]]. Each one has a prism which is a CRF polychoron, bringing the running total to 156. Some of these prisms are included in Klitzing's list of segmentotopes, but some are not because the corresponding Johnson solid is not orbiform (vertices lie on a glome).
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=== Richard Klitzing's segmentotopes ===
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The [[discovery index]] page serves as the authoritative list of D number assignments.
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Dr. Richard Klitzing enumerated the full set of 177 [[segmentochoron|segmentotopes]]: monostratic CRF polychora whose vertices lie on the surface of an inscribing [[glome]]. These can be divided into:
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*3 regular polychora ([[pyrochoron]], [[aerochoron]], [[geochoron]]);
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== Prismatoids ==
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*the [[3-pyrotomochoron]], a uniform polychoron;
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{{selfref|Total in this section (excluding stacks): 264}}
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*19 members of infinite families;
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*The first 64 CRF polychora are the [[List of uniform polychora|convex uniform polychora]], which can be divided up into 9 pyromorphs, 9 xylomorphs, 12 stauromorphs (not 15, because three were already covered as xylomorphs), 15 rhodomorphs, the [[snub demitesseract]] and the [[grand antiprism]], and 17 prisms of convex uniform polyhedra (not 18, because one was already covered as the tesseract, a stauromorph).
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*4 infinite families (Klitzing numbered these 174-177 however they are not individual polychora);
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*A further 92 CRF prisms are possible, based on each of the 92 [[Johnson solids]]. This brings the running total to 156. Some of these prisms are included in Klitzing's list of [[segmentotope]]s, but some are not due to their corresponding Johnson solids not being [[orbiform]] (having all vertices lie on a [[sphere]]).
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*all 17 prisms of the uniform polyhedra (not 18, because the cube prism is the geochoron, already counted above);
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*There are 30 [[CRFP4DP/Monostratic cupolic forms|CRF monostratic cupolic forms]] (or just ''cupolae''). These are constructed by placing uniform polyhedra from the same symmetry group (except in one special case, the [[snubdis antiprism]]) in each plane.
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*12 pyramids of CRF polyhedra;
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*There are 10 [[bicupolic ring]]s (not 12, because 2 are already counted).
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*25 prisms of [[orbiform]] Johnson solids;
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*There are 62 [[CRFP4DP/More prismatoids|more prismatoid forms]] (not 64, because 2 are already counted).
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*7 prismatoid forms (cupolae and antiprisms of regular polyhedra);
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*There are some [[CRFP4DP/Stacks|bistratic polychora formed by stacking monostratic polychora together]], but these have not been counted up yet.
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*all 9 bicupolic rings;
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*There are 6 [[ursachora]].
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*80 other polychora which are not listed elsewhere on this page. <span style='color: red'>some are, individually? this needs to be corrected</span>
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==== Prismatoid forms ====
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== Diminishings ==
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We can generate 52 CRF polychora by all possible combinations of {[[tetrahedron]], [[cube]], [[octahedron]], [[icosahedron]], [[square antiprism]], [[pentagonal antiprism]], [[triangular prism]], [[pentagonal prism]], [[square pyramid]], [[pentagonal pyramid]], [[diminished icosahedron]], [[metabidiminished icosahedron]], [[tridiminished icosahedron]]} × {pyramid, bipyramid, elongated pyramid, elongated bipyramid}. However, two of these - the "tetrahedral pyramid" and the "octahedral bipyramid" - are already covered as the [[pyrochoron]] and the [[aerochoron]] respectively, leaving us with 50 new CRF polychora. This brings the running total to 206.
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{{selfref|Main article: [[CRFP4DP/Diminishings]]}}
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*Some diminishings of uniform polychora produce various segmentotopes and CRF polychora.
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*There are many [[BT polychora]].
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*There are three '''known''' non-uniform convex scaliform (equilateral and vertex transitive) polytopes. All their ridges are regular.
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*There are 18 diminishings of the xylochoron (not 19, because one is the tesseract).
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*The tesseract augmentations are precisely those that are only ortho or para; all meta-diminishings are not augmented tesseracts; there are 6 of these.
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*The 600-cell has a large number of diminishings, two of which are uniform (the snub 24-cell and the grand antiprism). Removing icosahedral pyramids from the 600-cell generates a large number of CRF polychora; removing 24 in 24-cell configuration generates the snub 24-cell. Removing two rings of 10 vertices each from mutually complementary 2-planes generates the grand antiprism; removing subsets of these vertices generates various intermediates (full exploration of the possibilities still in progress).
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*There are some modified bisected 600-cells, lunae (wedge-like multiply-bisected 600-cells) and rotundae.
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The remaining CRF polyhedra cannot generate pyramidal forms for one (or both) of the following reasons:
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== Augmentations ==
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*the polyhedron's vertices are further from its center than its edge length, thus any pyramid of it would require base-apex edge lengths longer than base-base edge lengths, and thus not be CRF;
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{{selfref|Main article: [[CRFP4DP/Augmentations]]}}
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**note that this reason is implied if the polyhedron contains a [[contour]] with at least six edges, but the converse is not always true, e.g. in the case of the [[dodecahedron]]
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*The [[pyrochoron]] has a single augmentation, consisting of two pyrochora joined cell-to-cell.
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*the polyhedron cannot be [[inscribed in a sphere]], thus there is no point equidistant from all base points, thus any pyramid of it would have at least two different base-apex edge lengths, and thus not be CRF.
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*The [[tesseract]] has 14 augmentations with cubical pyramids, one of which corresponds with the [[xylochoron]]. However, these augmentations are also a subset of the xylochoron's ''diminishings'', so their count should not be included under the present category.
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*There are some modified tesseract augmentations and augmentations of truncated tesseracts.
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*There are probably many other augmentable uniform polychora, these have yet to be explored.
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*The duoprisms are a source of 1633 CRF polychora via augmentation with CRF pyramids, especially because the pentagonal prism pyramid is very shallow. This shallowness permits it to be fitted onto pentagonal prisms of n,5-duoprisms in various combinations up to n=20. The other prism pyramids (triangular and square) are less shallow, but still contribute a good number of CRF polychora.
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*Besides CRF pyramids, certain other segmentotopes can augment duoprisms to form CRFs. The full enumeration of such duoprism augmentations is currently in progress. Preliminary calculations indicate that augmentation with n-gonal magnabicupolic rings number in the millions, due to combinatorial explosion.
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[[wintersolstice]] originally proposed a list containing more polyhedra than those listed above, but this was incorrect due to the above reasons. He acknowledged that there was a mistake with the list some time ago, most likely realizing the same argument that has been written above, but did not give this explanation at the time.
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== Gyrations ==
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*wintersolstice discovered that [[CRFP4DP/Gyrations|some CRFs can be gyrated in various ways]].
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==== Cupolae of regular polyhedra ====
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== Modified Stott expansions ==
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We can generate 21 CRF polychora from the possible combinations of {tetrahedral, cubic, dodecahedral} × {cupola, orthobicupola, gyrobicupola, elongated cupola, elongated orthobicupola, elongated gyrobicupola, antiprism}. There are an additional 8 forms constructed as {octahedral, icosahedral} × {cupola, orthobicupola, elongated cupola, elongated orthobicupola}, as these forms do not use both duals. This gives 29 shapes in total.
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The elongated cubic orthobicupola is the same as the runcinated tesseract, leaving us with 28 new CRF polychora.
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Arguably, the CRFs described below could be classified as crown jewels, even though since their initial discovery they have been found in large numbers, which somewhat defeats the label "crown jewels".
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Each cupola is constructed as the spline from the base polyhedron to its [[peritruncate]]. In the case of gyrobicupolae, the "other end" of the polychoron is the dual of the base shape. In the case of antiprisms, the spline is directly from the base shape to its dual.
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=== Partial Stott-expansions ===
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{{selfref|Main article: [[Partial Stott-expansion]]}}
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Klitzing discovered in 2013 that some infinite families can be expanded according to a lower symmetry group, giving new polytopes. Expanding a 16-cell by a 4-fold subsymmetry, for example, produces a CRF polychoron best described as the convex hull of a tesseract and an octagon, both centered on the origin. This represents the first step in a series of partial Stott expansions that eventually yields the runcinated tesseract (x4o3o3x).
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<span style='color: red;'>The ability to construct these shapes with regular faces needs to be checked.</span>
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===EKF polytopes===
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In 2014 quickfur discovered a derivation of a bilbiro from an icosahedron. This led to the discovery of various partial expansions of the hydrochoron. This derivation is not merely a partial Stott expansion, but a ''modification'' of partial Stott expansion by adding an initial faceting step before the actual Stott expansion.  Student91 discovered an underlying general scheme where a polytope (usually uniform or regular) can be partially faceted according to some subsymmetry, and then Stott-expanded according to the same symmetry in order to restore convexity, often resulting in novel CRFs.  The faceting is done by "punning" a CD node label with an equivalent label that has a negative value, thereby producing a non-convex faceting, which is then restored to 0 by the subsequent Stott expansion.
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Two copies of the icosahedron-dodecahedron antiprism can be fitted together by their dodecahedral bases; the relative sizes of the icosahedron and dodecahedron of equal edge length ensures that the result is convex, and therefore CRF. It consists of 2 icosahedral cells, 100 tetrahedra, and 24 pentagonal pyramids. An elongated form is obtained by inserting a dodecahedral prism. Both forms have augmented and biaugmented variants. The other polyhedron-dual antiprisms do not produce convex CRFs this way, so this is a unique combination giving 6 new CRF polychora in total.
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In 3D, the icosahedron, for example, can be faceted according to a 2-fold subsymmetry and then expanded, producing a [[bilbiro]] (J91). The same process applied to a 3-fold subsymmetry produces the triangular hebesphenorotunda (J92). Applied to a 5-fold subsymmetry, this process produces J32, the pentagonal orthocupolarotunda.
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==== Bicupolic rings ====
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This process has come to be known by the acronym EKF (expando-kaleido-faceting). When applied to various 4D uniform polytopes, primarily those in the 120-cell family, the EKF operation produces a large number of new CRFs, many of them containing J32, J91, and/or J92 cells. A good number, probably the majority, of crown jewels in the D4.x numbering scheme are EKF polytopes.
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Nine CRF polychora are available from the possible combinations of {triangle, square, pentagon} × {ortho, gyro, magna}. [[Keiji]] discovered the ortho- and gyro- forms, and [[quickfur]] discovered the magna- form. Keiji has dubbed these shapes [[bicupolic ring]]s in general, and the specific naming pattern is ''n''-gonal ''form''bicupolic ring, e.g. ''square orthobicupolic ring''.
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The ortho- and gyro- forms are constructed as in [http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468&p=16497#p16461 this post]. The magna- forms are constructed as in [http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468&p=16497#p16493 this post] (second-to-last paragraph).
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These CRFs are included as ''wedges'' in Klitzing's list.
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==== Diminishings of uniform polychora ====
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Some diminishings of uniform polychora produce various segmentotopes.
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* The monodiminished rectified 5-cell is trigon||gyrated triangular prism (K4.6.2).
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* The bidiminished rectified 5-cell is line||orthogonal triangular prism (K4.8.2).
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* Deleting the vertices of an octahedron from the cantellated 5-cell produces cuboctahedron||truncated_tetrahedron (K4.48).
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* The rectified tesseract can be cut into three segmentochora: tetrahedron||truncated_tetrahedron, truncated_tetrahedron||truncated_dual_tetrahedron, and dual_truncated_tetrahedron||dual_tetrahedron.
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* The runcinated tesseract can be cut into two cube||rhombicuboctahedron and one rhombicuboctahedron prism. The former in turn can be cut into two square orthobicupolic rings and a square cupola prism, and the latter can be cut into two square cupola prisms and a 4,8-duoprism (octagonal prism prism).
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* The cantellated tesseract can be cut into rhombicuboctahedron||truncated_cube and a truncated cube prism.
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* The bisected 24-cell (see below) is the segmentochoron octahedron||cuboctahedron; it can be further diminished into square_pyramid||cuboctahedron (K4.31) and subsequently square||cuboctahedron (K4.28).
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==== Crown jewel ====
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An unusual segmentotope discovered by Klitzing is cube||icosahedron, consisting of 1 cube, 6 triangular prisms, 12 square pyramids, 8 tetrahedra, and 1 icosahedron.
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Marek found that there are exactly two ways that two copies of this segmentotope can be glued together at its icosahedral cell, each resulting in a CRF polychoron with 2 cubes, 12 triangular prisms, 24 square pyramids, and 16 tetrahedra.
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== Bistratic polychora ==
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Bistratic polychora are those whose vertices lie on 3 parallel hyperplanes.
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=== Monostratic stacks ===
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The simplest cases of bistratic polychora are those formed by stacking two monostratic polychora such that the result is CRF. This requires:
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# The two monostratic polychora have a common cell shape where the join will take place;
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# The stack of two monostratic polychora is convex, that is, the sum of dichoral angles between the joining cell and the lacing cells of the respective monostratics must be ≤ 180°.
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# Where the sum of dichoral angles is exactly 180°, an additional requirement is made that the result merged cell (from the combination of coplanar lacing cells) must be CRF, since there are some cases for which this is not true, such as joining together a square pyramid and a tetrahedron, which produces a sheared triangular prism with rhombus faces.
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=== Ursachora ===
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Wendy Krieger discovered a bistratic polychoron constructed from the convex hull of a unit tetrahedron, a tetrahedron scaled by the golden ratio, and a unit octahedron. It consists of 4 [[tridiminished icosahedra]] surrounding a tetrahedron, with 4 more tetrahedra on the other end connecting the tridiminished icosahedra to an octahedron. This shape is in fact a 4D analog of the tridiminished icosahedron itself, which has been nicknamed "Teddy" [http://teamikaria.com/hddb/forum/viewtopic.php?p=17756#p17756 by Klitzing] (from Jonathan Bowers' acronym for the '''t'''ri'''d'''iminished '''i'''cosahedron: '''t'''e'''d'''d'''i'''), a name which was quickly adopted by quickfur, although there is some confusion about the dimensions.
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Subsequently, a whole class of similar shapes have been discovered, all containing tridiminished icosahedra, which may be referred to as [[ursatope]]s (a pun on "teddy").
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So far, there are 6 known 4D CRF members of this family:
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* Tetrahedral ursachoron: xfo3oox3ooo&#xf in Wendy's lacing notation, consisting of 1+4=5 tetrahedron, 1 octahedron, 4 tridiminished icosahedra.
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* Octahedral ursachoron: xfo3oox4ooo&#xf, consisting of 1 octahedron, 1 cuboctahedron, 8 tridiminished icosahedra, 6 square pyramids.
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* Icosahedral ursachoron: xfo3oox5ooo&#xf, consisting of 1 icosahedron, 20 tridiminished icosahedra, 12 pentagonal pyramids and 1 icosidodecahedron. This shape happens to be a diminishing of the hemi-600-cell.
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* Tetrahedral expanded ursachoron: xfo3oox3xxx&#xf, consisting of 1 cuboctahedron, 1 truncated octahedron, 4 tridiminished icosahedra, 6 pentagonal prisms, 4 triangular prisms, 4 triangular cupola ''(to be verified)''.
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* Octahedral expanded ursachoron: xfo3oox4xxx&#xf, consisting of 1 rhombicuboctahedron, 1 truncated cube, 8 tridiminished icosahedra, 12 pentagonal prisms, 6 square cupola, 6 cubes.
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* Icosahedral expanded ursachoron: xfo3oox5xxx&#xf, consisting of 1 rhombicosidodecahedron, 1 truncated dodecahedron, 20 tridiminished icosahedra, 30+12 = 42 pentagonal prisms, 12 pentagonal cupola. ''(To be verified)''
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=== Birotundular ring ===
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Diminishing the 600-cell wedge #4 produces an analogue of the bicupolic rings: the pentagonal gyrobirotundular ring (see the section on 600-cell bisections). This CRF is bistratic, as its vertices can be placed on 3 parallel hyperplanes.
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== Convex non-uniform scaliform ==
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Outside the uniform polychora (uniformed-celled and vertex transitive), there are 4 '''known''' convex scaliform (equilateral and vertex transitive) polytopes. All their ridges are regular.
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;[[Bixylodiminished hydrochoron]] (aka ''bi-icositetradiminished 600-cell'')
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:The 600-cell can be made as the convex hull of five 24-cells. Removing the vertices of one and taking the convex hull creates the snub 24-cell. Removing another and taking the convex hull creates this shape. It cells are 48 tridimished icosahedra.
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;Prismatorhombato snub 24-cell
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:This is made by shrinking the 24 icosahedral cells of a snub 24-cell (all the tetrahedra are removed). Where icosahedra used to be joined by triangles they are now seperated by triangular prisms (96 in total). The holes in the shape are filled with 24 truncated tetrahedra and 96 triangular cupola.
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;Truncated tetrahedra cupoliprism (one of Richard Klitzing segmentotopes)
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:This is made by taking two truncated tetrahedra with hexagons lined up to triangles and then putting 8 triangular cuplolae on the hexagons and triangles, then filling in the holes with 5 tetrahedra.
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;Swirlprismatodiminished rectified 600-cell
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:The Rectified 600-cell can be made as the convex hull of 6 600-cells. Removing the vertices of one and taking the convex hull of the remaining vertices produces this shape. Its cells are 600 square pyramids, 120 pentagonal prisms and 120 pentagonal antiprisms.
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== Diminished polychora ==
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Some regular polychora can be diminished to give CRF polychora.
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=== Pyromorphs ===
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Some of the 5-cell family polychora can be diminished to give various segmentochora (included in Klitzing's list). Some diminishings are not monostratic, however:
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* The cantellated 5-cell can have two triangles deleted from it in such a way that it produces a polychoron with 2 hexagonal prisms, 4 square pyramids, 1 cuboctahedron, 4 triangular cupola, and 4 triangular prisms.
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* The runcitruncated 5-cell can have the vertices of a truncated tetrahedron deleted, producing a polychoron with 4 triangular cupola, 1 cuboctahedron, 6 hexagonal prisms, 4 triangular prisms, and 4 truncated tetrahedra.
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=== Stauromorphs ===
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* The diminished 16-cell coincides with the octahedral pyramid (see Prismatoid forms above).
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* The rectified tesseract can have the vertices of two non-antipodal, non-touching tetrahedra deleted, producing a polychoron with 2 truncated tetrahedra, 6 triangular cupola, 6 tetrahedra, and 1 cubocatahedron. (Metabidiminished rectified tesseract.) (Note that cutting off one tetrahedron or two antipodal tetrahedra produces various stacked segmentochora.)
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* The cantellated tesseract can have square magnabicupolic rings cut off from it to produce CRFs. Each candidate square magnabicupolic ring corresponds with one of the 24 ridges of a tesseract, and two cuttings are only allowed if these ridges do not share an edge. Thus, it corresponds with the number of non-adjacent vertices of the 24-cell; there are 19 such diminishings (see 24-cell diminishings). The octadiminished cantellated tesseract coincides with the 8,8-duoprism; many of the other diminishings correspond with augmentations of the 8,8-duoprism. Five of these diminishings do not correspond with any augmented 8,8-duoprism, because the bicupolic rings are cut off in a 'skewed' way that breaks duoprism symmetry.
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* The runcitruncated 16-cell can be diminished to form a rotunda (see below).
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=== Xylomorphs ===
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The 24-cell can have non-adjacent vertices deleted to form various CRF polychora (this is equivalent to cutting off cubical pyramids). There are 19 such diminishings, one of which is the tesseract. 13 of these coincide with augmentations of the tesseract with cubical pyramids.
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# Diminished 24-cell
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# Orthobidiminished 24-cell (ortho means two deleted vertices lie on antipodes of one octahedral cell)
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# Metabidiminished 24-cell (meta means two deleted vertices are separated by an additional edge past the antipodes of octahedral cells containing one of them)
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# Parabidiminished 24-cell (para means two deleted vertices are 24-cell antipodes)
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# Orthotridiminished 24-cell (all 3 vertices ortho to each other)
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# Paratridiminished 24-cell (two vertices are para to each other, ortho to the third)
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# Metatridiminished 24-cell (two vertices are meta, third is ortho to both)
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# Orthotetradiminished 24-cell (all 4 vertices ortho to each other)
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# Paratetradiminished 24-cell (one pair of vertices para to each other, ortho to other two)
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# Orthometatetradiminished 24-cell (3 vertices ortho to each other, last vertex meta to one of them)
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# Cyclotetradiminished 24-cell (4 vertices lie on a great circle)
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# Metametatetradiminished 24-cell (two pairs of ortho vertices, meta to each other)
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# Orthopentadiminished 24-cell (1 pair of para vertices, ortho to all others)
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# Metapentadiminished 24-cell (4 vertices ortho to each other, 5th vertex meta to all of them)
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# Parapentadiminished 24-cell (2 pairs of para vertices, all ortho to 5th)
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# Orthohexadiminished 24-cell (2 pairs of para vertices, ortho to each other & everything else)
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# Parahexadiminished 24-cell (3 pairs of para vertices)
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# Heptadiminished 24-cell (augmented tesseract)
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# (Octadiminished 24-cell == tesseract)
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The tesseract augmentations are precisely those that are only ortho or para; all meta-diminishings are not augmented tesseracts; there are 6 of these.
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The maximally-diminished 24-cells (those from which no more vertices can be deleted without becoming non-CRF) are the metametatetradiminished 24-cell, the metapentadiminished 24-cell, and the tesseract.
+
-
 
+
-
Furthermore, the 24-cell can be bisected to form a segmentochoron octahedron||cuboctahedron, which can be further diminished by deleting up to two more cubical pyramids. All three are segmentochora in Klitzing's list (see segmentotopes section above). Thus, there are 21 diminishings of the 24-cell altogether (not counting the tesseract). Among these, 18 are new CRFs not included in previous categories.
+
-
 
+
-
The 24-cell diminishings have been independently enumerated by wintersolstice, Marek, and quickfur.
+
-
 
+
-
=== Rhodomorphs ===
+
-
 
+
-
==== Diminished 600-cells ====
+
-
 
+
-
The 600-cell has a large number of diminishings, two of which are uniform (the snub 24-cell and the grand antiprism). Removing icosahedral pyramids from the 600-cell generates a large number of CRF polychora; removing 24 in 24-cell configuration generates the snub 24-cell. Removing two rings of 10 vertices each from mutually complementary 2-planes generates the grand antiprism; removing subsets of these vertices generates various intermediates (full exploration of the possibilities still in progress).
+
-
 
+
-
==== Bisected 600-cells ====
+
-
 
+
-
The 600-cell can be "bisected" by removing all vertices that lie on one side of an inscribed icosidodecahedron. (This is not a true bisection, as 12 edges that are bisected by the cutting hyperplane are completely removed from the result, instead of producing half-length edges, which would make the result non-CRF.) The result is a polychoron with 270 tetrahedra, 12 pentagonal pyramids, and 1 icosidodecahedron. It may be called a "hemi-600-cell". Various other diminishings past this point are possible, including Wendy's tristratic point||icosahedron||dodecahedron||icosidodecahedron, which is obtained by deleting the vertices of an inscribed icosahedron (larger than the edge length) from the bisected 600-cell; and a bistratic CRF having 1 icosahedron, 20 tridiminished icosahedra, 12 pentagonal pyramids, and 1 icosidodecahedron (see bistratics section above).
+
-
 
+
-
The hemi-600-cell can be further bisected by choosing the cutting hyperplane in such a way that it neatly bisects the bottom icosidodecahedron. Bisected edges are discarded and replaced with pentagonal pyramids to keep the result CRF. There are 4 such cuttings known, which result in wedges of various dichoral angles between the two resulting pentagonal rotundae:
+
-
* Wedge #1: 2 pentagonal rotundae, 12 pentagonal pyramids, and 210 tetrahedra. This is the widest wedge, with dichoral angle of 144° between the rotundae.
+
-
* Wedge #2: 2 pentagonal rotundae, 12 pentagonal pyramids, and 150 tetrahedra. A slightly narrower wedge, with dichoral angle of 108° between the rotundae.
+
-
* Wedge #3: 2 pentagonal rotundae, 12 pentagonal pyramids, and 90 tetrahedra. An even narrower wedge, with dichoral angle of 72° between rotundae.
+
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* Wedge #4: 2 pentagonal rotundae, 12 pentagonal pyramids, and 30 tetrahedra. The narrowest wedge, with dichoral angle of 36° between rotundae.
+
-
 
+
-
All of these wedges can be further diminished by deleting various vertices. Removing the vertex antipodal to the decagonal face in wedge #4 produces a rotunda analogue of the gyrobicupolic rings: the [[pentagonal gyrobirotundular ring]], having 1 pentagonal antiprism flanked by a ring of 20 tetrahedra, which in turn are flanked by a ring of 12 pentagonal pyramids, with two pentagonal rotundae closing up the shape.
+
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+
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=== Rotundae ===
+
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So far, five CRF rotundae have been discovered, all of which can be derived by diminishing uniform polychora. The vertices of these shapes lie in more than 2 parallel hyperplanes, therefore they are not included among Klitzing's segmentochora.
+
-
 
+
-
Mrrl discovered that a CRF polychoron can be cut from the rectified 120-cell when diminishing the latter. This polychoron consists of 1 icosidodecahedron, 12 pentagonal rotundae, and 40 tetrahedra. It can be considered the 4D analogue of the 3D pentagonal rotunda.
+
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+
-
A similar CRF rotunda can be obtained from the cantellated 600-cell by a similar cutting, producing a polychoron with 1 icosidodecahedron, 12 pentagonal rotundae, 42 pentagonal prisms, 20 cuboctahedra, and 20 triangular cupolae.
+
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+
-
Mrrl also found that the top of the second rotunda can be diminished, to obtain another CRF rotunda with 1 truncated icosahedron, 12 pentagonal rotunda, 30 pentagonal prisms, and 40 triangular cupolae.
+
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+
-
Two rotundae can be similarly obtained from the truncated 16-cell and the runcitruncated 16-cell, respectively. The first shape consists of 1 octahedron, 8 truncated tetrahedra, 6 square pyramids, and a truncated octahedron. The second shape consists of 1 (small) rhombicuboctahedron, 8 truncated tetrahedra, 12 hexagonal prisms, 6 square cupola, and 1 great rhombicuboctahedron. Both are CRF, but are not included in Klitzing's segmentochora because they have vertices that lie on 3 parallel hyperplanes.
+
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+
-
These rotundae have birotunda forms as well as their corresponding elongates, giving 20 new CRF polychora in total.
+
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+
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== Augmented uniform polychora ==
+
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+
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Some of the uniform polychora can be ''augmented'' (have CRF pyramids erected on one or more of their cells) and still remain CRF. The criteria for such augmentations to be CRF are:
+
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+
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* There must exist a CRF pyramid whose base is in the shape of the cell being augmented.
+
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* The sum of each dichoral angle of the pyramid's cells with its base and the dichoral angle of the cell being augmented and the corresponding neighbouring cell must be ≤ 180° in order to remain convex. If two adjacent cells are being augmented, then the sum of dichoral angles of two adjacent pyramid cells with their respective bases and the dichoral angle between the two augmented cells must be ≤ 180°.
+
-
* In the case where the sum of dichoral angles is exactly 180°, adjacent cells in the augments will merge; in such a case, the merged cells must themselves be CRF.
+
-
 
+
-
The [[pyrochoron]] has a single augmentation, consisting of two pyrochora joined cell-to-cell.
+
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+
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The [[tesseract]] has 14 augmentations with cubical pyramids, one of which corresponds with the [[xylochoron]]. However, these augmentations are also a subset of the xylochoron's ''diminishings'', so their count should not be included under the present category.
+
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+
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The tesseract (considered as a 4,4-duoprism) can be augmented with 4 square pyramid prisms and 4 line||square segmentochora to form a CRF polychoron having 4 cubes, 16 triangular prisms, and 16 tetrahedra. This is somewhat different augmentation from what is considered above, in that the initial augmentation creates a non-convex polychoron, but the gaps can be filled in with CRF segmentochora to form a valid CRF polychoron. This shape can also be generated as the convex hull of a tesseract and an octagon. It was discovered by [[User:Quickfur|quickfur]] on 9 Jan 2012.
+
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+
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<s>The [[xylochoron]] has 20 augmentations ([http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468&p=16905#p16905 forum post]).</s> <span style="color:red">This has been found to be invalid, due to it being based on a faulty computation of the xylochoron's dichoral angle.</span>
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+
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There are probably many other augmentable uniform polychora, these have yet to be explored.
+
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+
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=== Augmented duoprisms ===
+
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==== With pyramids ====
+
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The duoprisms are a source of 1633 CRF polychora via augmentation with CRF pyramids, especially because the pentagonal prism pyramid is very shallow. This shallowness permits it to be fitted onto pentagonal prisms of n,5-duoprisms in various combinations up to n=20. The other prism pyramids (triangular and square) are less shallow, but still contribute a good number of CRF polychora.
+
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+
-
The following lists the number of CRF polychora generated by augmenting duoprisms:
+
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+
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<table style="margin-left:auto;margin-right:auto;">
+
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<tr><th>3,3-duoprism: </th><td>3</td></tr>
+
-
<tr><th>3,4-duoprism: </th><td>5</td></tr>
+
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<tr><th>3,5-duoprism: </th><td>11</td></tr>
+
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<tr><th>3,6-duoprism: </th><td>4</td></tr>
+
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<tr><th>4,5-duoprism: </th><td>17</td></tr>
+
-
<tr><th>4,6-duoprism: </th><td>4</td></tr>
+
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<tr><th>4,7-duoprism: </th><td>4</td></tr>
+
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<tr><th>4,8-duoprism: </th><td>7</td></tr>
+
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<tr><th>5,5-duoprism: </th><td>35</td></tr>
+
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<tr><th>5,6-duoprism: </th><td>12</td></tr>
+
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<tr><th>5,7-duoprism: </th><td>17</td></tr>
+
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<tr><th>5,8-duoprism: </th><td>29</td></tr>
+
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<tr><th>5,9-duoprism: </th><td>45</td></tr>
+
-
<tr><th>5,10-duoprism: </th><td>77</td></tr>
+
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<tr><th>5,11-duoprism: </th><td>15</td></tr>
+
-
<tr><th>5,12-duoprism: </th><td>25</td></tr>
+
-
<tr><th>5,13-duoprism: </th><td>30</td></tr>
+
-
<tr><th>5,14-duoprism: </th><td>48</td></tr>
+
-
<tr><th>5,15-duoprism: </th><td>63</td></tr>
+
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<tr><th>5,16-duoprism: </th><td>98</td></tr>
+
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<tr><th>5,17-duoprism: </th><td>132</td></tr>
+
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<tr><th>5,18-duoprism: </th><td>208</td></tr>
+
-
<tr><th>5,19-duoprism: </th><td>290</td></tr>
+
-
<tr><th>5,20-duoprism: </th><td>454</td></tr>
+
-
<tr><th>Total: </th><td>1633 augmentations</td></tr>
+
-
</table>
+
-
 
+
-
The 4,4-duoprism is omitted here, because it coincides with the tesseract, the augmentations of which are covered under another category.
+
-
 
+
-
The sharp drop in the number of augmentations between the 3,5-duoprism and the 3,6-duoprism, between the 4,5-duoprism and the 4,6-duoprism, and between the 5,5-duoprism and the 5,6-duoprism is because pyramids of hexagonal (or higher) prisms cannot be CRF, since equilateral triangles tile the hexagon and so no hexagonal (or higher) pyramid can be formed without breaking the regular-faced requirement. Thus, only one of the duoprism's two rings can be augmented.
+
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+
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The drop between the 5,10-duoprism and the 5,11-duoprism is caused by the fact that adjacent pentagonal prism pyramids erected on an n-membered duoprism ring are no longer convex after n=10, so from the 5,11-duoprism onwards only non-adjacent augmentations are permitted, thus reducing the number of possible combinations. Adjacent augments on the 5,10-duoprism have pentagonal pyramid cells that are coplanar, thus merging into a pentagonal bipyramid.
+
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+
-
Augments of the 5,20-duoprism have pentagonal pyramids coplanar with the adjacent pentagonal prism, so they merge into elongated pentagonal pyramids. If the next prism in the ring is also augmented, then another pentagonal pyramid is added to the coplanar cell, turning it into an elongated pentagonal bipyramid.
+
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+
-
No other duoprisms can be augmented with CRF pyramids and still remain convex.
+
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+
-
==== With other segmentochora ====
+
-
 
+
-
Besides CRF pyramids, certain other segmentotopes can augment duoprisms to form CRFs. The full enumeration of such duoprism augmentations is currently in progress.
+
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+
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So far, it has been found that some of the n,6-duoprisms, n,8-duoprisms, and n,10-duoprisms can be augmented by the segmentotopes m-gon||2m-prism for m=3,4,5. These augments induce an ''orientation'' on both duoprism rings, reducing symmetry and increasing the number of distinct augmentations. These augmentations have been enumerated for the following duoprisms:
+
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+
-
* 3,6-duoprism: The first ring can be augmented with triangle||hexagonal_prism and the second ring with triangular prism pyramids, albeit not at the same time. Total: 9 augmentations.
+
-
* 4,6-duoprism: The first ring augmentable with triangle||hexagonal_prism; second ring with cubical pyramid, albeit not at the same time. Total: 7 augmentations.
+
-
* 5,6-duoprism: The first ring augmentable with triangle||hexagonal_prism; second ring with pentagonal prism pyramid. First ring augments reduce the symmetries of the second ring by half, thus increasing the number of distinct combinations. Total: 64 augmentations.
+
-
* 6,6-duoprism: First ring augmentable with triangle||hexagonal_prism; only one ring augmentable at a time. Total: 4 augmentations.
+
-
* 3,8-duoprism: First ring augmentable with square||octagonal_prism; second ring cannot be augmented. Total: 5 augmentations.
+
-
* 4,8-duoprism: First ring augmentable with square||octagonal prism; second ring with cubical pyramid (non-adjacent only). Both rings can be augmented simultaneously, but only when all augments on the first ring are in ''ortho'' orientation. Total: 44 augmentations.
+
-
* 5,8-duoprism: TBD.
+
-
* 8,8-duoprism: augmentable with square magnabicupolic rings; these are included as a subset of the diminishings and gyrations of the cantellated tesseract.
+
-
 
+
-
Since the n-gonal cupola may be constructed by radially expanding the triangular faces of the n-gonal pyramid, so the segmentochoron 2n-prism||n-gon (n-gonal magnabicupolic ring) may be constructed by radially expanding the pyramid cells of the n-prism pyramid. The dihedral angles of the triangles with the base polygon remain the same, implying that for every n-gonal prism pyramid, the n-gonal magnabicupolic ring exhibits the same dichoral angles between the base prism and the cupola and tetrahedral cells above it as the respective pyramids and tetrahedra in the n-gonal prism pyramid do. Therefore, the bicupolic rings may be fitted on m,2n-duoprisms in the corresponding positions as the prism pyramid augments occur on the m,n-duoprism.
+
-
 
+
-
Therefore, every augmentation of an m,n-duoprism by prism pyramids has a corresponding augmentation of the m,2n-duoprism with n-gonal magnabicupolic rings. So there are ''at least'' 1633 augmentations of duoprisms by these bicupolic rings. There are probably more, because these segmentochora may be rotated relative to each other in some of the m,2n-duoprisms, and for m=3,4,5, the other ring of prisms may be augmentable by m-gonal prism pyramids, leading to many more possible combinations.
+
-
 
+
-
== Other modified uniform polychora ==
+
-
 
+
-
wintersolstice discovered that some CRFs can be gyrated in various ways. For example, a cube||rhombicuboctahedron can have a square orthobicupolic ring cut off and replaced with a square gyrobicupolic ring.
+
-
 
+
-
The cantellated tesseract is a rich source of CRFs obtained by cutting off square magnabicupolic rings and gluing them back on with the "wrong" orientation (i.e., with the octagonal prism base rotated 45°). As a result, some octahedral cells are cut into square pyramids, and some rhombicuboctahedral cells are gyrated into pseudo-rhombicuboctahedra (elongated square gyrobicupola). Among the possibilies are:
+
-
* Paratetragyrated cantellated tesseract: 4 antipodal square magnabicupolic rings lying on two orthogonal great circles are gyrated, producing a polychoron with 8 pseudo-rhombicuboctahedra, and a number of square pyramids that meet with triangular prisms in an irregular way.
+
-
* Orthotetragyrated cantellated tesseract: 4 square magnabicupolic rings lying on the same great circle are gyrated. The result is a modified cantellated tesseract where 4 of the rhombicuboctahedral cells are "aligned wrongly" with the other 4.
+
-
* Two magnabicupolic rings lying on opposite ends of a single rhombicuboctahedron can be removed, and then one of them glued back onto the octagonal prism that remains from the original rhombicuboctahedral cell. This produces a fastigium-like CRF with 5-fold loop around one great circle (4 elongated square cupola + 1 rhombicuboctahedron), an odd deviation from tesseractic symmetry.
+
-
* The full set of possibilities has yet to be enumerated.
+
== Infinite families ==
== Infinite families ==
-
The obvious infinite family is that of the ''m'',''n''-duoprisms (''m'' ≥ ''n'' ≥ 3).
+
{{selfref|Main article: [[CRFP4DP/Infinite families]]}}
-
 
+
*The obvious infinite family is that of the ''m'',''n''-duoprisms (''m'' ≥ ''n'' ≥ 3).
-
There is also an infinite family of prisms of the ''n''-gonal antiprisms.
+
*There is also an infinite family of prisms of the ''n''-gonal antiprisms.
-
 
+
*[[Mrrl]] discovered an infinite family of ringed forms, with a 3-membered ring consisting of two antiprisms and a prism, with various Johnson polyhedra filling in the gaps. The first member contains two [[square antiprism]]s, one [[cube]], four [[tetrahedra]] and four [[square pyramid]]s. Details can be found in [http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468&p=16497#p16496 this post]. In general, members of this family consists of two ''n''-gonal antiprisms and an ''n''-gonal prism, forming a 3-membered ring, with ''n'' tetrahedra and ''n'' square pyramids filling in the lateral gaps, for all ''n'' ≥ 3. Keiji has devised a similar naming scheme to the one he used for the cupolic rings: the collective term is the family of ''biantiprismatic rings'', and the specific term is the ''n''-gonal biantiprismatic ring, e.g. ''[[square biantiprismatic ring]]''. These ringed forms are included as an infinite subfamily in Klitzing's list of segmentotopes where they are numbered among the ''wedges''.
-
[[Mrrl]] discovered an infinite family of ringed forms, with a 3-membered ring consisting of two antiprisms and a prism, with various Johnson polyhedra filling in the gaps. The first member contains two [[square antiprism]]s, one [[cube]], four [[tetrahedra]] and four [[square pyramid]]s. Details can be found in [http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468&p=16497#p16496 this post]. In general, members of this family consists of two ''n''-gonal antiprisms and an ''n''-gonal prism, forming a 3-membered ring, with ''n'' tetrahedra and ''n'' square pyramids filling in the lateral gaps, for all ''n'' ≥ 3. Keiji has devised a similar naming scheme to the one he used for the cupolic rings: the collective term is the family of ''biantiprismatic rings'', and the specific term is the ''n''-gonal biantiprismatic ring, e.g. ''square biantiprismatic ring''. These ringed forms are included as an infinite subfamily in Klitzing's list of segmentotopes where they are numbered among the ''wedges''.
+
*The n-gonal pyramid antiprisms (n-gonal pyramid || inverted gyro n-gonal pyramid) are CRF, and for n=4 and n=5, non-orbiform. (For n=3, it is identical to the 16-cell.) They are identical to the n-antiprism bipyramid.

Latest revision as of 05:05, 9 March 2019

See also: List of convex regular-faced polyhedra

This page documents an ongoing project to discover as many CRF polychora as possible, and perhaps as a long-term goal prove that every CRF polychoron has been found.

Other than infinite series, the non-Johnson CRF polyhedra are the regular polyhedra and the Archimedean polyhedra. The Johnson solids can be roughly divided into four categories: prismatoids (includes pyramids, cupolae, rotundae, and elongated/bi/gyro forms of the above), augmented polyhedra, diminished polyhedra and crown jewels. We will use the same categorizations here.

Richard Klitzing enumerated a list of 177 segmentochora, which are the orbiform CRF polychora. However, these overlap with several different categories below, so the segmentochora will not be considered a category of their own. TODO: find out how many segmentochora are not yet counted below.

Discovery index (D numbers)

As of February 2014, new CRF discoveries are assigned a discovery index (aka D number), as a way of uniquely identifying the discovery without committing to a specific categorization or naming of it, which may not be feasible due to insufficient information at the time of discovery.

The discovery index page serves as the authoritative list of D number assignments.

Prismatoids

Total in this section (excluding stacks): 264

Diminishings

Main article: CRFP4DP/Diminishings
  • Some diminishings of uniform polychora produce various segmentotopes and CRF polychora.
  • There are many BT polychora.
  • There are three known non-uniform convex scaliform (equilateral and vertex transitive) polytopes. All their ridges are regular.
  • There are 18 diminishings of the xylochoron (not 19, because one is the tesseract).
  • The tesseract augmentations are precisely those that are only ortho or para; all meta-diminishings are not augmented tesseracts; there are 6 of these.
  • The 600-cell has a large number of diminishings, two of which are uniform (the snub 24-cell and the grand antiprism). Removing icosahedral pyramids from the 600-cell generates a large number of CRF polychora; removing 24 in 24-cell configuration generates the snub 24-cell. Removing two rings of 10 vertices each from mutually complementary 2-planes generates the grand antiprism; removing subsets of these vertices generates various intermediates (full exploration of the possibilities still in progress).
  • There are some modified bisected 600-cells, lunae (wedge-like multiply-bisected 600-cells) and rotundae.

Augmentations

Main article: CRFP4DP/Augmentations
  • The pyrochoron has a single augmentation, consisting of two pyrochora joined cell-to-cell.
  • The tesseract has 14 augmentations with cubical pyramids, one of which corresponds with the xylochoron. However, these augmentations are also a subset of the xylochoron's diminishings, so their count should not be included under the present category.
  • There are some modified tesseract augmentations and augmentations of truncated tesseracts.
  • There are probably many other augmentable uniform polychora, these have yet to be explored.
  • The duoprisms are a source of 1633 CRF polychora via augmentation with CRF pyramids, especially because the pentagonal prism pyramid is very shallow. This shallowness permits it to be fitted onto pentagonal prisms of n,5-duoprisms in various combinations up to n=20. The other prism pyramids (triangular and square) are less shallow, but still contribute a good number of CRF polychora.
  • Besides CRF pyramids, certain other segmentotopes can augment duoprisms to form CRFs. The full enumeration of such duoprism augmentations is currently in progress. Preliminary calculations indicate that augmentation with n-gonal magnabicupolic rings number in the millions, due to combinatorial explosion.

Gyrations

Modified Stott expansions

Arguably, the CRFs described below could be classified as crown jewels, even though since their initial discovery they have been found in large numbers, which somewhat defeats the label "crown jewels".

Partial Stott-expansions

Main article: Partial Stott-expansion

Klitzing discovered in 2013 that some infinite families can be expanded according to a lower symmetry group, giving new polytopes. Expanding a 16-cell by a 4-fold subsymmetry, for example, produces a CRF polychoron best described as the convex hull of a tesseract and an octagon, both centered on the origin. This represents the first step in a series of partial Stott expansions that eventually yields the runcinated tesseract (x4o3o3x).

EKF polytopes

In 2014 quickfur discovered a derivation of a bilbiro from an icosahedron. This led to the discovery of various partial expansions of the hydrochoron. This derivation is not merely a partial Stott expansion, but a modification of partial Stott expansion by adding an initial faceting step before the actual Stott expansion. Student91 discovered an underlying general scheme where a polytope (usually uniform or regular) can be partially faceted according to some subsymmetry, and then Stott-expanded according to the same symmetry in order to restore convexity, often resulting in novel CRFs. The faceting is done by "punning" a CD node label with an equivalent label that has a negative value, thereby producing a non-convex faceting, which is then restored to 0 by the subsequent Stott expansion.

In 3D, the icosahedron, for example, can be faceted according to a 2-fold subsymmetry and then expanded, producing a bilbiro (J91). The same process applied to a 3-fold subsymmetry produces the triangular hebesphenorotunda (J92). Applied to a 5-fold subsymmetry, this process produces J32, the pentagonal orthocupolarotunda.

This process has come to be known by the acronym EKF (expando-kaleido-faceting). When applied to various 4D uniform polytopes, primarily those in the 120-cell family, the EKF operation produces a large number of new CRFs, many of them containing J32, J91, and/or J92 cells. A good number, probably the majority, of crown jewels in the D4.x numbering scheme are EKF polytopes.

Infinite families

Main article: CRFP4DP/Infinite families
  • The obvious infinite family is that of the m,n-duoprisms (m ≥ n ≥ 3).
  • There is also an infinite family of prisms of the n-gonal antiprisms.
  • Mrrl discovered an infinite family of ringed forms, with a 3-membered ring consisting of two antiprisms and a prism, with various Johnson polyhedra filling in the gaps. The first member contains two square antiprisms, one cube, four tetrahedra and four square pyramids. Details can be found in this post. In general, members of this family consists of two n-gonal antiprisms and an n-gonal prism, forming a 3-membered ring, with n tetrahedra and n square pyramids filling in the lateral gaps, for all n ≥ 3. Keiji has devised a similar naming scheme to the one he used for the cupolic rings: the collective term is the family of biantiprismatic rings, and the specific term is the n-gonal biantiprismatic ring, e.g. square biantiprismatic ring. These ringed forms are included as an infinite subfamily in Klitzing's list of segmentotopes where they are numbered among the wedges.
  • The n-gonal pyramid antiprisms (n-gonal pyramid || inverted gyro n-gonal pyramid) are CRF, and for n=4 and n=5, non-orbiform. (For n=3, it is identical to the 16-cell.) They are identical to the n-antiprism bipyramid.
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