Bicupolic ring (EntityClass, 16)
From Hi.gher. Space
(Difference between revisions)
(add summary and table) |
(add history section split from CRF polychora discovery project) |
||
| (7 intermediate revisions not shown) | |||
| Line 1: | Line 1: | ||
| - | <[#ontology [kind class] [cats Ringed_form | + | <[#ontology [kind class] [cats 4D Ringed_form Segmentotope]]> |
There are twelve '''bicupolic rings''' in [[Segmentochoron|Klitzing's list of segmentochora]], two of which turn out to be prisms. | There are twelve '''bicupolic rings''' in [[Segmentochoron|Klitzing's list of segmentochora]], two of which turn out to be prisms. | ||
| - | {|class='wikitable' | + | {|class='wikitable' style='text-align: center;' |
| - | ! | + | !style='width: 20%;'| |
| - | !Digonal | + | !style='width: 20%;'|Digonal |
| - | !Trigonal | + | !style='width: 20%;'|Trigonal |
| - | !Square | + | !style='width: 20%;'|Square |
| - | !Pentagonal | + | !style='width: 20%; padding-left: 5px; padding-right: 5px;'|Pentagonal |
|- | |- | ||
|ortho- | |ortho- | ||
| - | |[[Tetrahedral prism|K4.9]] | + | |style='vertical-align: bottom;'|<[#embed [hash 2Y6EC6FJBFXCPWWPHGZ9G626QE] [height 50]]><br/>[[Tetrahedral prism|K4.9]] |
| - | |[[Trigonal orthobicupolic ring|K4.25]] | + | |<[#embed [hash G1H1TK4VFZV38VZG5K66EQ0A4P] [height 50]]><br/>[[Trigonal orthobicupolic ring|K4.25]] |
| - | |[[Square orthobicupolic ring|K4.73]] | + | |<[#embed [hash RR35Y43QTQ6BY8FHP7E8M8PPEQ] [height 50]]><br/>[[Square orthobicupolic ring|K4.73]] |
| - | |[[Pentagonal orthobicupolic ring|K4.154]] | + | |<[#embed [hash ANKWCX42212YR7P9SGS7MPQEG2] [height 50]]><br/>[[Pentagonal orthobicupolic ring|K4.154]] |
|- | |- | ||
|gyro- | |gyro- | ||
| - | |[[Digonal gyrobicupolic ring|K4.8]] | + | |<[#embed [hash 2TCT0MYSHNJBPK594VWD8RYRSG] [height 50]]><br/>[[Digonal gyrobicupolic ring|K4.8]] |
| - | |[[Trigonal gyrobicupolic ring|K4.27]] | + | |<[#embed [hash N2BYGAZRMM4R665BCK6WTK7XT0] [height 50]]><br/>[[Trigonal gyrobicupolic ring|K4.27]] |
| - | |[[Square gyrobicupolic ring|K4.64]] | + | |<[#embed [hash 84ZQE0313240PWV0MBZWTCYHRV] [height 50]]><br/>[[Square gyrobicupolic ring|K4.64]] |
| - | |[[Pentagonal gyrobicupolic ring|K4.133]] | + | |<[#embed [hash EFF0R80EYX6GWM0ZN10BPD1FFW] [height 50]]><br/>[[Pentagonal gyrobicupolic ring|K4.133]] |
|- | |- | ||
|magna- | |magna- | ||
| - | |[[Square pyramid prism|K4.12]] | + | |style='vertical-align: bottom;'|<[#embed [hash 3NWJRDYM6HJXT8XWXVHRTK4HH2] [height 50]]><br/>[[Square pyramid prism|K4.12]] |
| - | |[[Trigonal magnabicupolic ring|K4.51]] | + | |<[#embed [hash PV974PFM6RMXYH518NV9TT7YG6] [height 50]]><br/>[[Trigonal magnabicupolic ring|K4.51]] |
| - | |[[Square magnabicupolic ring|K4.105]] | + | |<[#embed [hash WYC8PWYW55BDYQA32GE1AJYVH6] [height 50]]><br/>[[Square magnabicupolic ring|K4.105]] |
| - | |[[Pentagonal magnabicupolic ring|K4.165]] | + | |<[#embed [hash PGJPJECSKG22E2EEJEHJ4GBTZA] [height 50]]><br/>[[Pentagonal magnabicupolic ring|K4.165]] |
|} | |} | ||
| + | |||
| + | == History == | ||
| + | These CRFs were originally included as ''wedges'' in Klitzing's list. | ||
| + | |||
| + | [[Keiji]] independently rediscovered the ortho- and gyro- forms in late 2011, and [[quickfur]] rediscovered the magna- form shortly afterwards. 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]]''. [[student91]] noticed that Klitzing's K4.8 could be constructed as the digonal gyrobicupolic ring. | ||
| + | |||
| + | 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). | ||
| + | |||
| + | == Structure == | ||
| + | |||
| + | Each of these bicupolic rings consists of two n-gonal cupolae, a linking element, an opposite element corresponding with a Stott-expanded n-gon, and a ring of lateral cells that fill up the gaps in between. | ||
| + | * For the orthobicupolic rings, the linking element is an n-gonal prism, and the opposite element is a 2n-gon. The lateral cells consist of alternating tetrahedra and triangular prisms. | ||
| + | * For the gyrobicupolic rings, the linking element is an n-gonal antiprism, and the opposite element is a 2n-gon. The lateral cells consist of a ring of square pyramids in alternating orientations. | ||
| + | * For the magnabicupolic rings, the linking element is just an n-gon, and the opposite element is a 2n-gonal prism. The lateral cells consist of an alternating ring of triangular prisms and square pyramids. | ||
| + | |||
| + | In the digonal bicupolic rings, the digonal prism is simply the square, the digonal antiprism is the tetrahedron, and the 2n-gon is also the square (tetragon). Hence, the digonal cupola is just the triangular prism (the digon and the tetragon in parallel planes joined by a ring of alternating squares and triangles). | ||
Latest revision as of 16:43, 1 March 2014
There are twelve bicupolic rings in Klitzing's list of segmentochora, two of which turn out to be prisms.
| Digonal | Trigonal | Square | Pentagonal | |
|---|---|---|---|---|
| ortho- |  K4.9 |  K4.25 |  K4.73 |  K4.154 |
| gyro- |  K4.8 |  K4.27 |  K4.64 |  K4.133 |
| magna- |  K4.12 |  K4.51 |  K4.105 |  K4.165 |
History
These CRFs were originally included as wedges in Klitzing's list.
Keiji independently rediscovered the ortho- and gyro- forms in late 2011, and quickfur rediscovered the magna- form shortly afterwards. Keiji has dubbed these shapes bicupolic rings in general, and the specific naming pattern is n-gonal formbicupolic ring, e.g. square orthobicupolic ring. student91 noticed that Klitzing's K4.8 could be constructed as the digonal gyrobicupolic ring.
The ortho- and gyro- forms are constructed as in this post. The magna- forms are constructed as in this post (second-to-last paragraph).
Structure
Each of these bicupolic rings consists of two n-gonal cupolae, a linking element, an opposite element corresponding with a Stott-expanded n-gon, and a ring of lateral cells that fill up the gaps in between.
- For the orthobicupolic rings, the linking element is an n-gonal prism, and the opposite element is a 2n-gon. The lateral cells consist of alternating tetrahedra and triangular prisms.
- For the gyrobicupolic rings, the linking element is an n-gonal antiprism, and the opposite element is a 2n-gon. The lateral cells consist of a ring of square pyramids in alternating orientations.
- For the magnabicupolic rings, the linking element is just an n-gon, and the opposite element is a 2n-gonal prism. The lateral cells consist of an alternating ring of triangular prisms and square pyramids.
In the digonal bicupolic rings, the digonal prism is simply the square, the digonal antiprism is the tetrahedron, and the 2n-gon is also the square (tetragon). Hence, the digonal cupola is just the triangular prism (the digon and the tetragon in parallel planes joined by a ring of alternating squares and triangles).
