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WordNet

the most severe stage of a disease
the decisive moment in a novel or play; "the deathbed scene is the climax of the play" (同)culmination
the highest point of anything conceived of as growing or developing or unfolding; "the climax of the artists career"; "in the flood tide of his success" (同)flood_tide
arrangement of clauses in ascending order of forcefulness
the highest level or degree attainable; the highest stage of development; "his landscapes were deemed the acme of beauty"; "the artists gifts are at their acme"; "at the height of her career"; "the peak of perfection"; "summer was at its peak"; "...catapulted Einstein to the pinnacle of fame"; "the summit of his ambition"; "so many highest superlatives achieved by man"; "at the top of his profession" (同)height, elevation, peak, pinnacle, summit, superlative, meridian, tiptop, top

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〈C〉(物語・劇・行事などの)『細高潮』[『部』];(一般的に)絶頂[部] / 〈U〉漸層法(しだいに文勢を強調して最高潮に達する修辞法) / 絶頂に達する / …'を'絶頂に到達させる
《the ~》(…の)絶頂,極致(peak)《+『of』+『名』》

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/07/18 11:39:54」(JST)

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Uniform Triangular prism

Type	Prismatic uniform polyhedron
Elements	F = 5, E = 9 V = 6 (χ = 2)
Faces by sides	3{4}+2{3}
Schläfli symbol	t{2,3} or {3}x{}
Wythoff symbol	2 3 \| 2
Coxeter diagram
Symmetry group	D_3h, [3,2], (*322), order 12
Rotation group	D₃, [3,2]⁺, (322), order 6
References	U_76(a)
Dual	Triangular dipyramid
Properties	convex
Vertex figure 4.4.3

In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is oblique. A uniform triangular prism is a right triangular prism with equilateral bases, and square sides.

Equivalently, it is a polyhedron of which two faces are parallel, while the surface normals of the other three are in the same plane (which is not necessarily parallel to the base planes). These three faces are parallelograms. All cross-sections parallel to the base faces are the same triangle.

As a semiregular (or uniform) polyhedron

A right triangular prism is semiregular or, more generally, a uniform polyhedron if the base faces are equilateral triangles, and the other three faces are squares. It can be seen as a truncated trigonal hosohedron, represented by Schläfli symbol t{2,3}. Alternately it can be seen as the Cartesian product of a triangle and a line segment, and represented by the product {3}x{}. The dual of a triangular prism is a triangular bipyramid.

The symmetry group of a right 3-sided prism with triangular base is D_3h of order 12. The rotation group is D₃ of order 6. The symmetry group does not contain inversion.

Volume

The volume of any prism is the product of the area of the base and the distance between the two bases. In this case the base is a triangle so we simply need to compute the area of the triangle and multiply this by the length of the prism:

$V={\frac {1}{2}}bhl$ where b is the triangle base length, h is the triangle height, and l is the length between the triangles.

Truncated triangular prism

A truncated right triangular prism has one triangular face truncated (planed) at an oblique angle.^[1]

Facetings

There are two full D_2h symmetry facetings of a triangular prism, both with 6 isosceles triangle faces, one keeping the original top and bottom triangles, and one the original squares. Two lower C_3v symmetry faceting have one base triangle, 3 lateral crossed square faces, and 3 isosceles triangle lateral faces.

Convex	Facetings
D_3h symmetry			C_3v symmetry

2 {3} 3 {4}	3 {4} 6 ( ) v { }	2 {3} 6 ( ) v { }	1 {3} 3 t'{2} 6 ( ) v { }	1 {3} 3 t'{2} 3 ( ) v { }

Related polyhedra and tilings

Family of uniform prisms
Coxeter
Polyhedron
Tiling
Config.	3.4.4	4.4.4	5.4.4	6.4.4	7.4.4	8.4.4	9.4.4	10.4.4	11.4.4	12.4.4

Family of convex cupolae
n	2	3	4	5	6
Name	{2} \|\| t{2}	{3} \|\| t{3}	{4} \|\| t{4}	{5} \|\| t{5}	{6} \|\| t{6}
Cupola	Digonal cupola	Triangular cupola	Square cupola	Pentagonal cupola	Hexagonal cupola (Flat)
Related uniform polyhedra	Triangular prism	Cubocta- hedron	Rhombi- cubocta- hedron	Rhomb- icosidodeca- hedron	Rhombi- trihexagonal tiling

Symmetry mutations

This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.

*n32 symmetry mutation of truncated tilings: t{n,3}
Symmetry n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.	Noncompact hyperbolic
Symmetry n32 [n,3]	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]	[12i,3]	[9i,3]	[6i,3]
Truncated figures
Symbol	t{2,3}	t{3,3}	t{4,3}	t{5,3}	t{6,3}	t{7,3}	t{8,3}	t{∞,3}	t{12i,3}	t{9i,3}	t{6i,3}
Triakis figures
Config.	V3.4.4	V3.6.6	V3.8.8	V3.10.10	V3.12.12	V3.14.14	V3.16.16	V3.∞.∞

This polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.

*n32 symmetry mutation of expanded tilings: 3.4.n.4
Symmetry n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paracomp.
Symmetry n32 [n,3]	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]
Figure
Config.	3.4.2.4	3.4.3.4	3.4.4.4	3.4.5.4	3.4.6.4	3.4.7.4	3.4.8.4	3.4.∞.4

Compounds

There are 4 uniform compounds of triangular prisms:

Compound of four triangular prisms, compound of eight triangular prisms, compound of ten triangular prisms, compound of twenty triangular prisms.

Honeycombs

There are 9 uniform honeycombs that include triangular prism cells:

Gyroelongated alternated cubic honeycomb, elongated alternated cubic honeycomb, gyrated triangular prismatic honeycomb, snub square prismatic honeycomb, triangular prismatic honeycomb, triangular-hexagonal prismatic honeycomb, truncated hexagonal prismatic honeycomb, rhombitriangular-hexagonal prismatic honeycomb, snub triangular-hexagonal prismatic honeycomb, elongated triangular prismatic honeycomb

Related polytopes

The triangular prism is first in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (equilateral triangles and squares in the case of the triangular prism). In Coxeter's notation the triangular prism is given the symbol −1₂₁.

k₂₁ figures in n dimensional
Space	Finite						Euclidean	Hyperbolic
E_n	3	4	5	6	7	8	9	10
Coxeter group	E₃=A₂A₁	E₄=A₄	E₅=D₅	E₆	E₇	E₈	E₉ = E ~ 8 {\displaystyle {\tilde {E}}_{8}} = E₈⁺	E₁₀ = T ¯ 8 {\displaystyle {\bar {T}}_{8}} = E₈⁺⁺
Coxeter diagram
Symmetry	[3^−1,2,1]	[3^0,2,1]	[3^1,2,1]	[3^2,2,1]	[3^3,2,1]	[3^4,2,1]	[3^5,2,1]	[3^6,2,1]
Order	12	120	192	51,840	2,903,040	696,729,600	∞
Graph							-	-
Name	−1₂₁	0₂₁	1₂₁	2₂₁	3₂₁	4₂₁	5₂₁	6₂₁

Four dimensional space

The triangular prism exists as cells of a number of four-dimensional uniform 4-polytopes, including:

Four dimensional polytopes with triangular prisms
tetrahedral prism	octahedral prism	cuboctahedral prism	icosahedral prism	icosidodecahedral prism	Truncated dodecahedral prism

Rhombi-cosidodecahedral prism	Rhombi-cuboctahedral prism	Truncated cubic prism	Snub dodecahedral prism	n-gonal antiprismatic prism

Cantellated 5-cell	Cantitruncated 5-cell	Runcinated 5-cell	Runcitruncated 5-cell	Cantellated tesseract	Cantitruncated tesseract	Runcinated tesseract	Runcitruncated tesseract

Cantellated 24-cell	Cantitruncated 24-cell	Runcinated 24-cell	Runcitruncated 24-cell	Cantellated 120-cell	Cantitruncated 120-cell	Runcinated 120-cell	Runcitruncated 120-cell

References

^ William F. Kern, James R Bland,Solid Mensuration with proofs, 1938, p.81

Weisstein, Eric W., "Triangular prism", MathWorld.
Interactive Polyhedron: Triangular Prism
surface area and volume of a triangular prism

English Journal

Pollen morphology of Vochysiaceae tree species in the State of Santa Catarina, Southern Brazil.

Barth OM, Pinto Da Luz CF.
Revista de biología tropical.Rev Biol Trop.2014 Sep;62(3):1209-15.
Tropical Vochysiaceae includes mainly trees, and also shrubs and subshrubs. Three genera and seven species are present in the Brazilian state of Santa Catarina. The pollen morphology of six species of trees, belonging to three genera of the Vochysiaceae A. St-Hil. family, was studied. Herbaria sampl
PMID 25412545

An endoscopic, cadaveric analysis of the roof of the fourth ventricle.

Salma A1, Yeremeyeva E, Baidya NB, Sayers MP, Ammirati M.
Journal of clinical neuroscience : official journal of the Neurosurgical Society of Australasia.J Clin Neurosci.2013 May;20(5):710-4. doi: 10.1016/j.jocn.2012.05.051. Epub 2013 Mar 16.
We performed endoscopic dissections of the roof of the fourth ventricle in eight fresh human cadaveric heads to characterize the endoscopic anatomy of the roof of the fourth ventricle and the anatomical configuration of the structures forming its roof. We also made three-dimensional (3D) silicone ca
PMID 23507044

Researches on regenerative medicine-current state and prospect.

Wang ZG1, Xiao K.
Chinese journal of traumatology = Zhonghua chuang shang za zhi / Chinese Medical Association.Chin J Traumatol.2012;15(5):259-67.
Since 1980s, the rapid development of tissue engineering and stem cell research has pushed regenerative medicine to a new fastigium, and regenerative medicine has become a noticeable research field in the international biology and medicine. In China, about 100 million patients need repair and regene
PMID 23069095

Japanese Journal

消化器管ノ形態及ビ運動機能ニ關スル研究其ノ3 發熱ト胃竝ニ腸管運動機能トノ關係

本多稔
岡山医学会雑誌 45(12), 2927-2943, 1933-12-31
… Was die Motilität des Dünn-und Dickdarmes anbelangt, so wurde sie beim Fastigium des Fiebers deutlich langsam, beim Decrementi aber im Gegenteil beschleunigt. …
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實驗的發熱時ニ於ケル網状織内被細胞系統機能及ビ白血球像ニ就テ

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岡山医学会雑誌 44(1), 182-197, 1932-01-31
… In unseren Untersuchungen über die Funktion des Reticuloendothelialsystems mittels der Eisenzuckerinjektion und zugleich auch über das Leukozytenbild der infolge von Typhusvaccin-oder Thermininjektion, sowie auch von Wärmestich fiebernden Kaninchen wurden folgende Resultate erzielt: 1) Bei Vaccin- und Therminfieber zeigte sich eine mehr oder weniger gesteigerte Funktion des betreffenden Systems sowohl im Stadium incrementi als auch im Fastigium, im Stadium decrementi dagegen trat eine deutliche Abschwächung ein. …
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