Pavare trihexagonală romboidală

Pavare trihexagonală romboidală
Descriere
Tippavare semiregulată
Configurația fețeiV3.4.6.4
Simbol Wythoff3 | 6 2
Simbol Schläflirr{6,3} sau
Diagramă Coxeter
Grup de simetriep6m, [6,3], (*632)
Grup de rotațiep6, [6,3]+, (632)
Poliedru dualpavare rombitrihexagonală
Proprietățitranzitivă pe fețe

În geometrie pavarea trihexagonală romboidală este o duală a pavărilor semiregulate cunoscute sub numele de pavări rombitrihexagonale. Laturile pavări pot fi formate prin suprapunerea intersecțiilor pavării triunghiulare și ale celei hexagonale regulate. Fiecare față romboidală a acestei pavări are unghiurile de 120°, 90°, 60° și 90°. Este una dintre cele opt pavări ale planului în care fiecare latură se află pe o dreaptă de simetrie a pavărilor.[1][2]

Poliedre și pavări înrudite

Duala: pavare rombitrihexagonală

Este una dintre cele 7 pavări uniforme duale în simetrie hexagonală, inclusiv dualele regulate.

Pavări hexagonale/triughiulare uniforme duale
Simetrie: [6,3], (*632) [6,3]+, (632)
V63 V3.122 V(3.6)2 V36 V3.4.6.4 V.4.6.12 V34.6

Această pavare are variante tranzitive pe fețe, care pot deforma romboizii în trapeze sau patrulatere mai generale. Ignorând culorile feței de mai jos, simetria completă este p6m, iar simetria inferioară este p31m, cu 3 plane de oglindire care se întâlnesc într-un punct, și puncte de rotație cu trei poziții.[3]

variante izoedrice
Simetrie p6m, [6,3], (*632) p31m, [6,3+], (3*3)
Formă
Fețe Romboizi Jumătăți de hexagon regulat Patrulatere
Pavarea trihexagonală romboidală suprapusă peste pavarea trihexagonală

Această pavare este legată de pavarea trihexagonală prin divizarea triunghiurilor și hexagoanelor în triunghiuri și asamblarea triunghiurilor învecinate în romboedre.

Pavarea trihexagonală romboidală face parte dintr-un set de pavări duale uniforme, corespunzătoare dualelor pavărior rombitrihexagonale.

Variante de simetrie

Această pavare este legată topologic de secvența de pavări cu configurațiile fețelor V3.4.n.4 și continuă cu pavările planului hiperbolic. Aceste figuri tranzitive pe fețe au simetria în notația orbifold (*n32).

Variante de pavări expandate duale cu simetrie *n32: V3.4.n.4
Simetrie
*n32
[n,3]
Sferice Euclid. Hiperb. compacte Paracomp.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Config.
feței

V3.4.2.4

V3.4.3.4

V3.4.4.4

V3.4.5.4

V3.4.6.4

V3.4.7.4

V3.4.8.4

V3.4.∞.4

Alte pavări romboidale

Sunt posibile și alte pavări romboidale.

Simetria față de centru permite ca planul să fie umplut de romboizi cu dimensiuni crescătoare, sau cu o topologie ca a pavării pătrate, V4.4.4.4. Mai jos este un exemplu cu simetrie hexagonală diedrică.

Simetrie D6, [6], (*66) pmg, [∞,(2,∞)+], (22*) p6m, [6,3], (*632)
Pavare
Configurație V4.4.4.4 V6.4.3.4

Note

  1. ^ en Kirby, Matthew; Umble, Ronald (), „Edge tessellations and stamp folding puzzles”, Mathematics Magazine, 84 (4): 283–289, arXiv:0908.3257Accesibil gratuit, doi:10.4169/math.mag.84.4.283, MR 2843659 
  2. ^ en Eric W. Weisstein, Dual tessellation la MathWorld. (See comparative overlay of this tiling and its dual)
  3. ^ Tilings and Patterns

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