diminished trapezohedron
{{short description|Polyhedron made by truncating one end of a trapezohedron}}
{{Infobox polyhedron
| image = Diminished square trapezohedron.png
| caption = Example square form
| type =
| euler =
| faces = {{mvar|n}} kites
{{mvar|n}} triangles
1 {{nowrap|{{mvar|n}}-gon}}
| edges = {{math|4n}}
| vertices = {{math|2n + 1}}
| schläfli =
| wythoff =
| coxeter =
| symmetry = {{math|Cyclic symmetry, [n], (*nn)}}
| rotation_group = {{math|C{{sub|n}}, [n]{{sup|+}}, (nn)}}
| surface_area =
| volume =
| angle =
| dual = self-dual
| properties = convex
}}
In geometry, a diminished trapezohedron is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular {{nowrap|{{mvar|n}}-gonal}} base face, {{mvar|n}} triangle faces around the base, and {{mvar|n}} kites meeting on top. The kites can also be replaced by rhombi with specific proportions.
Along with the set of pyramids and elongated pyramids, these figures are topologically self-dual.
It can also be seen as an augmented {{nowrap|{{mvar|n}}-gonal}} antiprism, with a {{nowrap|{{mvar|n}}-gonal}} pyramid augmented onto one of the {{nowrap|{{mvar|n}}-gonal}} faces, and whose height is adjusted so the upper antiprism triangle faces can be made coparallel to the pyramid faces and merged into kite-shaped faces.
They're also related to the gyroelongated pyramids, as augmented antiprisms and which are Johnson solids for {{math|1=n = 4, 5}}. This sequence has sets of two triangles instead of kite faces.
Examples
| class="wikitable"
|+ Diminished trapezohedra |
| Symmetry
!C3v !C4v !C5v !C6v !C7v !C8v ... |
|---|
| align=center
!Image | 80px | 80px | 80px | 80px | 80px |
| align=center
!Rhombic | 80px | 80px | 80px | 80px | 80px | 80px |
| align=center
!Net | 80px | 80px | 80px | 80px | 80px | 80px |
| align=center valign=top
!Faces | 3 trapezoids | 4 trapezoids | 5 trapezoids | 6 trapezoids | 7 trapezoids | 8 trapezoids |
| align=center
!Edges |12 |16 |20 |24 |28 |32 |
| align=center
!Vertices |7 |9 |11 |13 |15 |17 |
| colspan=8|Trapezohedra |
| align=center
!Symmetry !D3d !D4d !D5d !D6d !D7d !D8d |
| align=center
!Image | | |
| align=center
!Faces | 3+3 rhombi | 4+4 kites | 5+5 kites | 6+6 kites | 7+7 kites | |
| align=center
!Edges |12 |16 |20 |24 |28 | |
| align=center
!Vertices |8 |10 |12 |14 |16 | |
| colspan=8|Gyroelongated pyramid or (augmented antiprisms) |
| align=center
!Symmetry !C3v !C4v !C5v !C6v !C7v !C8v |
| align=center
!Image |100px |
| align=center
!Faces |9+1 triangles |12 triangles |15 triangles |18 triangles | |
Special cases
There are three special case geometries of the diminished trigonal trapezohedron. The simplest is a diminished cube. The Chestahedron, named after artist Frank Chester, is constructed with equilateral triangles around the base, and the geometry adjusted so the kite faces have the same area as the equilateral triangles.{{Cite web|url=http://frankchester.com/chestahedron-geometry/|title=Chestahedron Geometry|website=The Art & Science of Frank Chester|language=en-US|access-date=2020-01-22}}{{Cite web|url=https://demonstrations.wolfram.com/TransformingATetrahedronIntoAChestahedron/|title=Transforming a Tetrahedron into a Chestahedron|last=Donke|first=Hans-Joakim|date=March 2011|website=Wolfram Alpha|url-status=live|access-date=22 January 2020|archive-url=https://web.archive.org/web/20141007023603/http://demonstrations.wolfram.com/TransformingATetrahedronIntoAChestahedron/ |archive-date=2014-10-07 }} The last can be seen by augmenting a regular tetrahedron and an octahedron, leaving 10 equilateral triangle faces, and then merging 3 sets of coparallel equilateral triangular faces into 3 (60 degree) rhombic faces. It can also be seen as a tetrahedron with 3 of 4 of its vertices rectified. The three rhombic faces fold out flat to form half of a hexagram.
| class=wikitable width=640
|+ Diminished trigonal trapezohedron variations |
| Heptahedron topology #31 Diminished cube !Chestahedron !Augmented octahedron |
|---|
| align=center |
| align=center |
| align=center
|3 squares |3 kite faces |3 60 degree rhombic faces |
See also
References
{{reflist}}
- [http://dmccooey.com/polyhedra/SymmetricSelfDuals.html Symmetries of Canonical Self-Dual Polyhedra] 7F,C3v:[http://dmccooey.com/polyhedra/SelfDualHeptahedron2.html] 9,C4v:[http://dmccooey.com/polyhedra/SelfDualEnneahedron2.html] 11,C5v:[http://dmccooey.com/polyhedra/SelfDualHendecahedron2.html], 13,C6v:[http://dmccooey.com/polyhedra/SelfDualTridecahedron2.html], 15,C7v:[http://dmccooey.com/polyhedra/SelfDualPentadecahedron2.html].
{{Polyhedron navigator}}