Prismă dodecagonală

Descriere
Prismă dodecagonală uniformă

(model 3D)
Tip	poliedru uniform, U_76j
Fețe	14 (2 dodecagoane regulate 12 pătrate,)
Laturi (muchii)	36
Vârfuri	24
χ	2
Configurația vârfului	4.4.12
Simbol Wythoff	2 12 \| 2 2 2 6 \|
Simbol Schläfli	t{2,12} sau {12}×{}
Diagramă Coxeter
Grup de simetrie	D_12h, [12,2], (*12.2.2), ordin 48
Grup de rotație	D₁₂, [12,2]⁺, (12.2.2), ordin 24
Arie	$A=2A_{b}+A_{lat}$
Volum	$V=A_{b}h$
Poliedru dual	bipiramidă dodecagonală
Proprietăți	convexă
Figura vârfului

În geometrie prisma dodecagonală este o prismă cu baza dodecagonală. Are 14 fețe, 36 de laturi și 24 de vârfuri.^[1] Deoarece are 14 fețe, în principiu este un tetradecaedru.

Prisma dodecagonală uniformă are indicele de poliedru uniform U_76j.^[2]

Ca poliedru semiregulat (sau uniform)

Dacă fețele sunt toate regulate, prisma dodecagonală este un poliedru semiregulat, mai general, un poliedru uniform, fiind a zecea într-un set infinit de prisme formate din fețe laterale pătrate și două baze poligoane regulate. Poate fi văzută ca un hosoedru dodecagonal trunchiat, reprezentat de simbolul Schläfli t{2,12}. Alternativ, poate fi văzută ca produsul cartezian al unui dodecagon regulat și al unui segment, și reprezentat prin produsul {12}×{}. Dualul unei prisme decagonale este o bipiramidă dodecagonală.

Simetrie

Grupul de simetrie al unei prisme dodecagonale drepte este D_12h de ordinul 48. Grupul de rotație este D₁₂ de ordinul 24.

Formule

Ca la toate prismele, aria totală $A$ este de două ori aria bazei ( $A b$ ) plus aria laterală, iar volumul $V$ este produsul dintre aria bazei și înălțimea (distanța dintre planele celor două baze) $h$ .

Pentru o prismă cu baza dodecagonală regulată cu latura $a$ , aria $A$ are formula:^[3]

A=2\,A_{b}+12\,ah=6\cot {\frac {\pi }{12}}\,a^{2}+12\,ah\approx 22,392305\,a^{2}+12\,ah

Pentru $a$ = 1 și $h$ = 1 aria este ≈ 34,392305.

Formula volumului $V$ este:^[3]

V=A_{b}h=3\cot {\frac {\pi }{12}}\,a^{2}h

Pentru $a$ = 1 și $h$ = 1 volumul este ≈ 11,196152.

Utilizări

Este folosită în construcția a doi faguri uniformi prismatici:

Fagure prismatic triunghiular-hexagonal	Fagure prismatic hexagonal trunchiat

Noua monedă britanică de 1 liră (£1), care a intrat în circulație în martie 2017, are forma unei prisme dodecagonale.^[4]

Poliedre înrudite

Familia prismelor n-gonale uniforme
Denumirea prismei	Prismă digonală	Prismă triunghiulară	Prismă tetragonală	Prismă pentagonală	Prismă hexagonală	Prismă heptagonală	Prismă octogonală	Prismă eneagonală	Prismă decagonală	Prismă endecagonală	Prismă dodecagonală	...	Prismă apeirogonală
Imagine												...
Pavare sferică												Pavare plană
Config. vârfului	2.4.4	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	...	∞.4.4
Diagramă Coxeter												...

Note

^ en Pugh, Anthony (1976), Polyhedra: A Visual Approach, University of California Press, pp. 21, 27, 62, ISBN 9780520030565 .
^ en Eric W. Weisstein, Uniform Polyhedron la MathWorld.
^ ^a ^b de Regelmäßiges Prisma - Rechner, rechneronline.de, accesat 2022-08-24
^ en „New 12-sided pound coin to enter circulation in March”. BBC News. 1 ianuarie 2017. Accesat în 20 aprilie 2017.

Vezi și

Lista poliedrelor uniforme

Legături externe

en Eric W. Weisstein, Prism la MathWorld.

Portal matematică

en Klitzing, Richard. „3D uniform polyhedra”. Cheie: twip

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