Lista de sólidos de Johnson

En geometría, los poliedros son objetos tridimensionales en los que los puntos están conectados por líneas para formar polígonos . Los puntos, líneas y polígonos de un poliedro se denominan vértices , aristas y caras , respectivamente. ^[1] Se considera que un poliedro es convexo si: ^[2]

• El camino más corto entre dos de sus vértices se encuentra en su interior o en su límite .
• Ninguna de sus caras es coplanar , es decir, no comparten el mismo plano y no quedan “planas”.
• Ninguno de sus bordes es colineal , es decir, no son segmentos de la misma línea.

Un poliedro convexo cuyas caras son polígonos regulares se conoce como sólido de Johnson o, a veces, como sólido de Johnson-Zalgaller . Algunos autores excluyen los poliedros uniformes de la definición. Un poliedro uniforme es un poliedro en el que las caras son regulares y son isogonales ; los ejemplos incluyen sólidos platónicos y arquimedianos , así como prismas y antiprismas . ^[3] Los sólidos de Johnson reciben su nombre del matemático estadounidense Norman Johnson (1930-2017), quien publicó una lista de 92 poliedros de este tipo en 1966. Su conjetura de que la lista estaba completa y no existían otros ejemplos fue demostrada por el matemático ruso-israelí Victor Zalgaller (1920-2020) en 1969. ^[4]

Algunos de los sólidos de Johnson pueden clasificarse como poliedros elementales , lo que significa que no pueden separarse por un plano para crear dos pequeños poliedros convexos con caras regulares. Los sólidos de Johnson que satisfacen este criterio son los seis primeros: pirámide cuadrada equilátera , pirámide pentagonal , cúpula triangular , cúpula cuadrada , cúpula pentagonal y rotonda pentagonal . El criterio también lo satisfacen otros once sólidos de Johnson, específicamente el icosaedro tridisminuido , el rombicosidodecaedro parabidisminuido , el rombicosidodecaedro tridisminuido , el disfenoide romo , el antiprisma cuadrado romo , la esfenocorona , la esfenomegacorona , la hebesfenomegacorona , el disfenocínculo , la bilunabirotonda y la hebesfenorrotonda triangular . ^[5] El resto de los sólidos de Johnson no son elementales, y se construyen utilizando los primeros seis sólidos de Johnson junto con sólidos platónicos y arquimedianos en varios procesos. La aumentación implica unir los sólidos de Johnson a una o más caras de poliedros, mientras que la elongación o giroelongación implican unirlos a las bases de un prisma o antiprisma, respectivamente. Algunos otros se construyen por disminución , la eliminación de uno de los primeros seis sólidos de una o más de las caras de un poliedro. ^[6]

La siguiente tabla contiene los 92 sólidos de Johnson, con longitud de arista . La tabla incluye la enumeración del sólido (denotada como ). ^[7] También incluye el número de vértices, aristas y caras de cada sólido, así como su grupo de simetría , área de superficie y volumen . Cada poliedro tiene sus propias características , incluyendo simetría y medida. Se dice que un objeto tiene simetría si hay una transformación que lo asigna a sí mismo. Todas esas transformaciones pueden estar compuestas en un grupo , junto con el número de elementos del grupo , conocido como orden . En el espacio bidimensional, estas transformaciones incluyen rotar alrededor del centro de un polígono y reflejar un objeto alrededor de la bisectriz perpendicular de un polígono. Un polígono que se rota simétricamente por se denota por , un grupo cíclico de orden ; combinando esto con la simetría de reflexión da como resultado la simetría del grupo diedro de orden . ^[8] En los grupos de puntos de simetría tridimensionales , las transformaciones que preservan la simetría de un poliedro incluyen la rotación alrededor de la línea que pasa por el centro de la base, conocida como eje de simetría , y la reflexión relativa a los planos perpendiculares que pasan por la bisectriz de una base, que se conoce como simetría piramidal de orden . La transformación que preserva la simetría de un poliedro reflejándolo a través de un plano horizontal se conoce como simetría prismática de orden . La simetría antiprismática de orden preserva la simetría rotando su mitad inferior y reflejándose a través del plano horizontal. ^[9] El grupo de simetría de orden preserva la simetría mediante rotación alrededor del eje de simetría y reflexión en el plano horizontal; el caso específico que preserva la simetría mediante una rotación completa es de orden 2, a menudo denotado como . ^[10] La medición de los poliedros incluye el área de la superficie y el volumen . Un área es una medida bidimensional calculada por el producto de la longitud y el ancho; Para un poliedro, el área de la superficie es la suma de las áreas de todas sus caras. ^[11] Un volumen es una medida de una región en el espacio tridimensional. ^[12]${\estilo de visualización a}$$Estilo de visualización J_{n}$${\estilo de visualización A}$${\estilo de visualización V}$${\textstyle {\frac {360^{\circ }}{n}}}$$Estilo de visualización C_{n}$${\estilo de visualización n}$ $Estilo de visualización D_{n}$${\estilo de visualización 2n}$ ${\displaystyle C_{n\mathrm {v}}}$${\estilo de visualización 2n}$ ${\displaystyle D_{n\mathrm {h}}}$${\estilo de visualización 4n}$ ${\displaystyle D_{n\mathrm {d}}}$${\estilo de visualización 4n}$${\displaystyle C_{n\mathrm {h}}}$${\estilo de visualización 2n}$${\displaystyle C_{1\mathrm {h}}}$${\displaystyle C_{s}}$El volumen de un poliedro se puede determinar de diferentes maneras: ya sea a través de su base y altura (como para las pirámides y los prismas ), cortándolo en pedazos y sumando sus volúmenes individuales, o encontrando la raíz de un polinomio que represente al poliedro. ^[13]

Los 92 sólidos de Johnson

$Estilo de visualización J_{n}$	Nombre solido	Imagen	Vértices	Bordes	Caras	Grupo de simetría y su orden [14]	Área de superficie y volumen [15]
1	Pirámide cuadrada equilátera		5	8	5	$Estilo de visualización C_{4v}$de orden 8	${\displaystyle {\begin{aligned}A&=\left(1+{\sqrt {3}}\right)a^{2}\\&\aproximadamente 2,7321a^{2}\\V&={\frac {\sqrt {2}}{6}}a^{3}\\&\aproximadamente 0,2357a^{3}\end{aligned}}}$
2	Pirámide pentagonal		6	10	6	$Estilo de visualización C_{5v}$de orden 10	${\displaystyle {\begin{aligned}A&={\frac {a^{2}}{2}}{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\\&\aproximadamente 3,8855a^{2}\\V&=\left({\frac {5+{\sqrt {5}}}{24}}\right)a^{3}\\&\aproximadamente 0,3015a^{3}\end{aligned}}}$
3	Cúpula triangular		9	15	8	$Estilo de visualización C3v$de orden 6	${\displaystyle {\begin{aligned}A&=\left(3+{\frac {5{\sqrt {3}}}{2}}\right)a^{2}\\&\approx 7.3301a^{2}\\V&=\left({\frac {5}{3{\sqrt {2}}}}\right)a^{3}\\&\approx 1.1785a^{3}\end{aligned}}}$
4	Cúpula cuadrada		12	20	10	${\displaystyle C_{4v}}$de orden 8	${\displaystyle {\begin{aligned}A&=\left(7+2{\sqrt {2}}+{\sqrt {3}}\right)a^{2}\\&\approx 11.5605a^{2}\\V&=\left(1+{\frac {2{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 1.9428a^{3}\end{aligned}}}$
5	Cúpula pentagonal		15	25	12	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&=\left({\frac {1}{4}}\left(20+5{\sqrt {3}}+{\sqrt {5\left(145+62{\sqrt {5}}\right)}}\right)\right)a^{2}\\&\approx 16.5798a^{2}\\V&=\left({\frac {1}{6}}\left(5+4{\sqrt {5}}\right)\right)a^{3}\\&\approx 2.3241a^{3}\end{aligned}}}$
6	Rotonda pentagonal		20	35	17	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&=\left({\frac {1}{2}}\left(5{\sqrt {3}}+{\sqrt {10\left(65+29{\sqrt {5}}\right)}}\right)\right)a^{2}\\&\approx 22.3472a^{2}\\V&=\left({\frac {1}{12}}\left(45+17{\sqrt {5}}\right)\right)a^{3}\\&\approx 6.9178a^{3}\end{aligned}}}$
7	Pirámide triangular alargada		7	12	7	${\displaystyle C_{3v}}$de orden 6	${\displaystyle {\begin{aligned}A&=\left(3+{\sqrt {3}}\right)a^{2}\\&\approx 4.7321a^{2}\\V&=\left({\frac {1}{12}}\left({\sqrt {2}}+3{\sqrt {3}}\right)\right)a^{3}\\&\approx 0.5509a^{3}\end{aligned}}}$
8	Pirámide cuadrada alargada		9	16	9	${\displaystyle C_{4v}}$de orden 8	${\displaystyle {\begin{aligned}A&=\left(5+{\sqrt {3}}\right)a^{2}\\&\approx 6.7321a^{2}\\V&=\left(1+{\frac {\sqrt {2}}{6}}\right)a^{3}\\&\approx 1.2357a^{3}\end{aligned}}}$
9	Pirámide pentagonal alargada		11	20	11	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&={\frac {20+5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}a^{2}\\&\approx 8.8855a^{2}\\V&=\left({\frac {5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}}{24}}\right)a^{3}\\&\approx 2.022a^{3}\end{aligned}}}$
10	Pirámide cuadrada giroelongada		9	20	13	${\displaystyle C_{4v}}$de orden 8	${\displaystyle {\begin{aligned}A&=(1+3{\sqrt {3}})a^{2}\\&\approx 6.1962a^{2}\\V&={\frac {1}{6}}\left({\sqrt {2}}+2{\sqrt {4+3{\sqrt {2}}}}\right)a^{3}\\&\approx 1.1927a^{3}\end{aligned}}}$
11	Pirámide pentagonal giroelongada		11	25	16	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(15{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 8.2157a^{2}\\V&={\frac {1}{24}}\left(25+9{\sqrt {5}}\right)a^{3}\\&\approx 1.8802a^{3}\end{aligned}}}$
12	Bipirámide triangular		5	9	6	${\displaystyle D_{3h}}$de orden 12	${\displaystyle {\begin{aligned}A&={\frac {3{\sqrt {3}}}{2}}a^{2}\\&\approx 2.5981a^{2}\\V&={\frac {\sqrt {2}}{6}}a^{3}\\&\approx 0.2358a^{3}\end{aligned}}}$
13	Bipirámide pentagonal		7	15	10	${\displaystyle D_{5h}}$de orden 20	${\displaystyle {\begin{aligned}A&={\frac {5{\sqrt {3}}}{2}}a^{2}\\&\approx 4.3301a^{2}\\V&={\frac {1}{12}}\left(5+{\sqrt {5}}\right)a^{3}\\&\approx 0.603a^{3}\end{aligned}}}$
14	Bipirámide triangular alargada		8	15	9	${\displaystyle D_{3h}}$de orden 12	${\displaystyle {\begin{aligned}A&={\frac {3}{2}}\left(2+{\sqrt {3}}\right)a^{2}\\&\approx 5.5981a^{2}\\V&={\frac {1}{12}}\left(2{\sqrt {2}}+3{\sqrt {3}}\right)a^{3}\\&\approx 0.6687a^{3}\end{aligned}}}$
15	Bipirámide cuadrada alargada		10	20	12	${\displaystyle D_{4h}}$de orden 16	${\displaystyle {\begin{aligned}A&=2\left(2+{\sqrt {3}}\right)a^{2}\\&\approx 7.4641a^{2}\\V&={\frac {1}{3}}\left(3+{\sqrt {2}}\right)a^{3}\\&\approx 1.4714a^{3}\end{aligned}}}$
16	Bipirámide pentagonal alargada		12	25	15	${\displaystyle D_{5h}}$de orden 20	${\displaystyle {\begin{aligned}A&={\frac {5}{2}}\left(2+{\sqrt {3}}\right)a^{2}\\&\approx 9.3301a^{2}\\V&={\frac {1}{12}}\left(5+{\sqrt {5}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{3}\\&\approx 2.3235a^{3}\end{aligned}}}$
17	Bipirámide cuadrada giroelongada		10	24	16	${\displaystyle D_{4d}}$de orden 16	${\displaystyle {\begin{aligned}A&=4{\sqrt {3}}a^{2}\\&\approx 6.9282a^{2}\\V&={\frac {1}{3}}\left({\sqrt {2}}+{\sqrt {4+3{\sqrt {2}}}}\right)a^{3}\\&\approx 1.4284a^{3}\end{aligned}}}$
18	Cúpula triangular alargada		15	27	14	${\displaystyle C_{3v}}$de orden 6	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(18+5{\sqrt {3}}\right)a^{2}\\&\approx 13.3301a^{2}\\V&={\frac {1}{6}}\left(5{\sqrt {2}}+9{\sqrt {3}}\right)a^{3}\\&\approx 3.7766a^{3}\end{aligned}}}$
19	Cúpula cuadrada alargada		20	36	18	${\displaystyle C_{4v}}$de orden 8	${\displaystyle {\begin{aligned}A&=(15+2{\sqrt {2}}+{\sqrt {3}})a^{2}\\&\approx 19.5605a^{2}\\V&=\left(3+{\frac {8{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 6.7712a^{3}\end{aligned}}}$
20	Cúpula pentagonal alargada		25	45	22	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(60+5{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 26.5798a^{2}\\V&={\frac {1}{6}}\left(5+4{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 10.0183a^{3}\end{aligned}}}$
21	Rotonda pentagonal alargada		30	55	27	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}a^{2}\left(20+5{\sqrt {3}}+5{\sqrt {5+2{\sqrt {5}}}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)\\&\approx 32.3472a^{2}\\V&={\frac {1}{12}}a^{3}\left(45+17{\sqrt {5}}+30{\sqrt {5+2{\sqrt {5}}}}\right)\\&\approx 14.612a^{3}\end{aligned}}}$
22	Cúpula triangular giroelongada		15	33	20	${\displaystyle C_{3v}}$de orden 6	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(6+11{\sqrt {3}}\right)a^{2}\\&\approx 12.5263a^{2}\\V&={\frac {1}{3}}{\sqrt {{\frac {61}{2}}+18{\sqrt {3}}+30{\sqrt {1+{\sqrt {3}}}}}}a^{3}\\&\approx 3.5161a^{3}\end{aligned}}}$
23	Cúpula cuadrada giroelongada		20	44	26	${\displaystyle C_{4v}}$de orden 8	${\displaystyle {\begin{aligned}A&=(7+2{\sqrt {2}}+5{\sqrt {3}})a^{2}\\&\approx 18.4887a^{2}\\V&=\left(1+{\frac {2}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}\right)a^{3}\\&\approx 6.2108a^{3}\end{aligned}}}$
24	Cúpula pentagonal giroelongada		25	55	32	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(20+25{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 25.2400a^{2}\\V&=\left({\frac {5}{6}}+{\frac {2}{3}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 9.0733a^{3}\end{aligned}}}$
25	Rotonda pentagonal giroelongada		30	65	37	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(15{\sqrt {3}}+\left(5+3{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}\right)a^{2}\\&\approx 31.0075a^{2}\\V&=\left({\frac {45}{12}}+{\frac {17}{12}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 13.6671a^{3}\end{aligned}}}$
26	Girobifastigium		8	14	8	${\displaystyle D_{2d}}$de orden 8	${\displaystyle {\begin{aligned}A&=\left(4+{\sqrt {3}}\right)a^{2}\\&\approx 5.7321a^{2}\\V&=\left({\frac {\sqrt {3}}{2}}\right)a^{3}\\&\approx 0.866a^{3}\end{aligned}}}$
27	Ortobicúpula triangular		12	24	14	${\displaystyle D_{3h}}$de orden 12	${\displaystyle {\begin{aligned}A&=2\left(3+{\sqrt {3}}\right)a^{2}\\&\approx 9.4641a^{2}\\V&={\frac {5{\sqrt {2}}}{3}}a^{3}\\&\approx 2.357a^{3}\end{aligned}}}$
28	Ortobicúpula cuadrada		16	32	18	${\displaystyle D_{4h}}$de orden 16	${\displaystyle {\begin{aligned}A&=2(5+{\sqrt {3}})a^{2}\\&\approx 13.4641a^{2}\\V&=\left(2+{\frac {4{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 3.8856a^{3}\end{aligned}}}$
29	Girobicúpula cuadrada		16	32	18	${\displaystyle D_{4d}}$de orden 16	${\displaystyle {\begin{aligned}A&=2(5+{\sqrt {3}})a^{2}\\&\approx 13.4641a^{2}\\V&=\left(2+{\frac {4{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 3.8856a^{3}\end{aligned}}}$
30	Ortobicúpula pentagonal		20	40	22	${\displaystyle D_{5h}}$de orden 20	${\displaystyle {\begin{aligned}A&=\left(10+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 17.7711a^{2}\\V&={\frac {1}{3}}\left(5+4{\sqrt {5}}\right)a^{3}\\&\approx 4.6481a^{3}\end{aligned}}}$
31	Girobicúpula pentagonal		20	40	22	${\displaystyle D_{5d}}$de orden 20	${\displaystyle {\begin{aligned}A&=\left(10+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 17.7711a^{2}\\V&={\frac {1}{3}}\left(5+4{\sqrt {5}}\right)a^{3}\\&\approx 4.6481a^{3}\end{aligned}}}$
32	Ortocupularrotonda pentagonal		25	50	27	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&=\left(5+{\frac {1}{4}}{\sqrt {1900+490{\sqrt {5}}+210{\sqrt {75+30{\sqrt {5}}}}}}\right)a^{2}\\&\approx 23.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 9.2418a^{3}\end{aligned}}}$
33	Rotonda girocupular pentagonal		25	50	27	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&=\left(5+{\frac {15}{4}}{\sqrt {3}}+{\frac {7}{4}}{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 23.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 9.2418a^{3}\end{aligned}}}$
34	Ortobirotonda pentagonal		30	60	32	${\displaystyle D_{5h}}$de orden 20	${\displaystyle {\begin{aligned}A&=\left((5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 29.306a^{2}\\V&={\frac {1}{6}}(45+17{\sqrt {5}})a^{3}\\&\approx 13.8355a^{3}\end{aligned}}}$
35	Ortobicúpula triangular alargada		18	36	20	${\displaystyle D_{3h}}$de orden 12	${\displaystyle {\begin{aligned}A&=2(6+{\sqrt {3}})a^{2}\\&\approx 15.4641a^{2}\\V&=\left({\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\\&\approx 4.9551a^{3}\end{aligned}}}$
36	Girobicúpula triangular alargada		18	36	20	${\displaystyle D_{3d}}$de orden 12	${\displaystyle {\begin{aligned}A&=2(6+{\sqrt {3}})a^{2}\\&\approx 15.4641a^{2}\\V&=\left({\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\\&\approx 4.9551a^{3}\end{aligned}}}$
37	Girobicúpula cuadrada alargada		24	48	26	${\displaystyle D_{4d}}$de orden 16	${\displaystyle {\begin{aligned}A&=2(9+{\sqrt {3}})a^{2}\\&\approx 21.4641a^{2}\\V&=\left(4+{\frac {10{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 8.714a^{3}\end{aligned}}}$
38	Ortobicúpula pentagonal alargada		30	60	32	${\displaystyle D_{5h}}$de orden 20	${\displaystyle {\begin{aligned}A&=\left(20+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 27.7711a^{2}\\V&={\frac {1}{6}}\left(10+8{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 12.3423a^{3}\end{aligned}}}$
39	Girobicúpula pentagonal alargada		30	60	32	${\displaystyle D_{5d}}$de orden 20	${\displaystyle {\begin{aligned}A&=\left(20+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 27.7711a^{2}\\V&={\frac {1}{6}}\left(10+8{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 12.3423a^{3}\end{aligned}}}$
40	Ortocupularotunda pentagonal alargada		35	70	37	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(60+{\sqrt {10\left(190+49{\sqrt {5}}+21{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 33.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}+6{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 16.936a^{3}\end{aligned}}}$
41	Rotonda girocupular pentagonal alargada		35	70	37	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(60+{\sqrt {10\left(190+49{\sqrt {5}}+21{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 33.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}+6{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 16.936a^{3}\end{aligned}}}$
42	Ortobirotonda pentagonal alargada		40	80	42	${\displaystyle D_{5h}}$de orden 20	${\displaystyle {\begin{aligned}A&=\left(10+{\sqrt {30\left(10+3{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 39.306a^{2}\\V&={\frac {1}{6}}\left(45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 21.5297a^{3}\end{aligned}}}$
43	Girobirotunda pentagonal alargada		40	80	42	${\displaystyle D_{5d}}$de orden 20	${\displaystyle {\begin{aligned}A&=\left(10+{\sqrt {30\left(10+3{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 39.306a^{2}\\V&={\frac {1}{6}}\left(45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 21.5297a^{3}\end{aligned}}}$
44	Bicúpula triangular giroelongada		18	42	26	${\displaystyle D_{3}}$de orden 6	${\displaystyle {\begin{aligned}A&=\left(6+5{\sqrt {3}}\right)a^{2}\\&\approx 14.6603a^{2}\\V&={\sqrt {2}}\left({\frac {5}{3}}+{\sqrt {1+{\sqrt {3}}}}\right)a^{3}\\&\approx 4.6946a^{3}\end{aligned}}}$
45	Bicúpula cuadrada giroelongada		24	56	34	${\displaystyle D_{4}}$de orden 8	${\displaystyle {\begin{aligned}A&=\left(10+6{\sqrt {3}}\right)a^{2}\\&\approx 20.3923a^{2}\\V&=\left(2+{\frac {4}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}\right)a^{3}\\&\approx 8.1536a^{3}\end{aligned}}}$
46	Bicúpula pentagonal giroelongada		30	70	42	${\displaystyle D_{5}}$de orden 10	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(20+15{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 26.4313a^{2}\\V&=\left({\frac {5}{3}}+{\frac {4}{3}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 11.3974a^{3}\end{aligned}}}$
47	Cupularotunda pentagonal giroelongada		35	80	47	${\displaystyle C_{5}}$de orden 5	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(20+35{\sqrt {3}}+7{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 32.1988a^{2}\\V&=\left({\frac {55}{12}}+{\frac {25}{12}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 15.9911a^{3}\end{aligned}}}$
48	Birotunda pentagonal giroelongada		40	90	52	${\displaystyle D_{5}}$de orden 10	${\displaystyle {\begin{aligned}A&=\left(10{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 37.9662a^{2}\\V&=\left({\frac {45}{6}}+{\frac {17}{6}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 20.5848a^{3}\end{aligned}}}$
49	Prisma triangular aumentado		7	13	8	${\displaystyle C_{2v}}$de orden 4	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}(4+3{\sqrt {3}})a^{2}\\&\approx 4.5981a^{2}\\V&={\frac {1}{12}}(2{\sqrt {2}}+3{\sqrt {3}})a^{3}\\&\approx 0.6687a^{3}\end{aligned}}}$
50	Prisma triangular biaumentado		8	17	11	${\displaystyle C_{2v}}$de orden 4	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}(2+5{\sqrt {3}})a^{2}\\&\approx 5.3301a^{2}\\V&={\sqrt {{\frac {59}{144}}+{\frac {1}{\sqrt {6}}}}}a^{3}\\&\approx 0.9044a^{3}\end{aligned}}}$
51	Prisma triangular triaumentado		9	21	14	${\displaystyle D_{3h}}$de orden 12	${\displaystyle {\begin{aligned}A&={\frac {7{\sqrt {3}}}{2}}a^{2}\\&\approx 6.0622a^{2}\\V&={\frac {2{\sqrt {2}}+{\sqrt {3}}}{4}}a^{3}\\&\approx 1.1401a^{3}\end{aligned}}}$
52	Prisma pentagonal aumentado		11	19	10	${\displaystyle C_{2v}}$de orden 4	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(8+2{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 9.173a^{2}\\V&={\frac {1}{12}}{\sqrt {233+90{\sqrt {5}}+12{\sqrt {50+20{\sqrt {5}}}}}}a^{3}\\&\approx 1.9562a^{3}\end{aligned}}}$
53	Prisma pentagonal biaumentado		12	23	13	${\displaystyle C_{2v}}$de orden 4	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}a^{2}\left(6+4{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)\\&\approx 9.9051a^{2}\\V&={\frac {1}{12}}a^{3}{\sqrt {257+90{\sqrt {5}}+24{\sqrt {50+20{\sqrt {5}}}}}}\\&\approx 2.1919a^{3}\end{aligned}}}$
54	Prisma hexagonal aumentado		13	22	11	${\displaystyle C_{2v}}$de orden 4	${\displaystyle {\begin{aligned}A&=(5+4{\sqrt {3}})a^{2}\\&\approx 11.9282a^{2}\\V&={\frac {1}{6}}\left({\sqrt {2}}+9{\sqrt {3}}\right)a^{3}\\&\approx 2.8338a^{3}\end{aligned}}}$
55	Prisma hexagonal parabiaumentado		14	26	14	${\displaystyle D_{2h}}$de orden 8	${\displaystyle {\begin{aligned}A&=(4+5{\sqrt {3}})a^{2}\\&\approx 12.6603a^{2}\\V&={\frac {1}{6}}\left(2{\sqrt {2}}+9{\sqrt {3}}\right)a^{3}\\&\approx 3.0695a^{3}\end{aligned}}}$
56	Prisma hexagonal metabiaumentado		14	26	14	${\displaystyle C_{2v}}$de orden 4	${\displaystyle {\begin{aligned}A&=(4+5{\sqrt {3}})a^{2}\\&\approx 12.6603a^{2}\\V&={\frac {1}{6}}\left(2{\sqrt {2}}+9{\sqrt {3}}\right)a^{3}\\&\approx 3.0695a^{3}\end{aligned}}}$
57	Prisma hexagonal triaumentado		15	30	17	${\displaystyle D_{3h}}$de orden 12	${\displaystyle {\begin{aligned}A&=3\left(1+2{\sqrt {3}}\right)a^{2}\\&\approx 13.3923a^{2}\\V&=\left({\frac {1}{\sqrt {2}}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\\&\approx 3.3052a^{3}\end{aligned}}}$
58	Dodecaedro aumentado		21	35	16	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(5{\sqrt {3}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 21.0903a^{2}\\V&={\frac {1}{24}}\left(95+43{\sqrt {5}}\right)a^{3}\\&\approx 7.9646a^{3}\end{aligned}}}$
59	Dodecaedro parabiaumentado		22	40	20	${\displaystyle D_{5d}}$de orden 20	${\displaystyle {\begin{aligned}A&={\frac {5}{2}}\left({\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 21.5349a^{2}\\V&={\frac {1}{6}}\left(25+11{\sqrt {5}}\right)a^{3}\\&\approx 8.2661a^{3}\end{aligned}}}$
60	Dodecaedro metabiaumentado		22	40	20	${\displaystyle C_{2v}}$de orden 4	${\displaystyle {\begin{aligned}A&={\frac {5}{2}}\left({\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 21.5349a^{2}\\V&={\frac {1}{6}}\left(25+11{\sqrt {5}}\right)a^{3}\\&\approx 8.2661a^{3}\end{aligned}}}$
61	Dodecaedro triaumentado		23	45	24	${\displaystyle C_{3v}}$de orden 6	${\displaystyle {\begin{aligned}A&={\frac {3}{4}}\left(5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 21.9795a^{2}\\V&={\frac {5}{8}}\left(7+3{\sqrt {5}}\right)a^{3}\\&\approx 8.5676a^{3}\end{aligned}}}$
62	Icosaedro metabidisminuido		10	20	12	${\displaystyle C_{2v}}$de orden 4	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(5{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 7.7711a^{2}\\V&={\frac {1}{6}}\left(5+2{\sqrt {5}}\right)a^{3}\\&\approx 1.5787a^{3}\end{aligned}}}$
63	Icosaedro tri-disminuido		9	15	8	${\displaystyle C_{3v}}$de orden 6	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 7.3265a^{2}\\V&=\left({\frac {5}{8}}+{\frac {7{\sqrt {5}}}{24}}\right)a^{3}\\&\approx 1.2772a^{3}\end{aligned}}}$
64	Icosaedro tridisminuido aumentado		10	18	10	${\displaystyle C_{3v}}$de orden 6	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(7{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 8.1925a^{2}\\V&={\frac {1}{24}}\left(15+2{\sqrt {2}}+7{\sqrt {5}}\right)a^{3}\\&\approx 1.395a^{3}\end{aligned}}}$
65	Tetraedro truncado aumentado		15	27	14	${\displaystyle C_{3v}}$de orden 6	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(6+13{\sqrt {3}}\right)a^{2}\\&\approx 14.2583a^{2}\\V&={\frac {11}{2{\sqrt {2}}}}a^{3}\\&\approx 3.8891a^{3}\end{aligned}}}$
66	Cubo truncado aumentado		28	48	22	${\displaystyle C_{4v}}$de orden 8	${\displaystyle {\begin{aligned}A&=(15+10{\sqrt {2}}+3{\sqrt {3}})a^{2}\\&\approx 34.3383a^{2}\\V&=\left(8+{\frac {16{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 15.5425a^{3}\end{aligned}}}$
67	Cubo truncado biaumentado		32	60	30	${\displaystyle D_{4h}}$de orden 16	${\displaystyle {\begin{aligned}A&=2\left(9+4{\sqrt {2}}+2{\sqrt {3}}\right)a^{2}\\&\approx 36.2419a^{2}\\V&=(9+6{\sqrt {2}})a^{3}\\&\approx 17.4853a^{3}\end{aligned}}}$
68	Dodecaedro truncado aumentado		65	105	42	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(20+25{\sqrt {3}}+110{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 102.1821a^{2}\\V&=\left({\frac {505}{12}}+{\frac {81{\sqrt {5}}}{4}}\right)a^{3}\\&\approx 87.3637a^{3}\end{aligned}}}$
69	Dodecaedro truncado parabiaumentado		70	120	52	${\displaystyle D_{5d}}$de orden 20	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(20+15{\sqrt {3}}+50{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 103.3734a^{2}\\V&={\frac {1}{12}}\left(515+251{\sqrt {5}}\right)a^{3}\\&\approx 89.6878a^{3}\end{aligned}}}$
70	Dodecaedro truncado metabiaumentado		70	120	52	${\displaystyle C_{2v}}$de orden 4	${\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(20+15{\sqrt {3}}+50{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 103.3734a^{2}\\V&={\frac {1}{12}}\left(515+251{\sqrt {5}}\right)a^{3}\\&\approx 89.6878a^{3}\end{aligned}}}$
71	Dodecaedro truncado triaumentado		75	135	62	${\displaystyle C_{3v}}$de orden 6	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(60+35{\sqrt {3}}+90{\sqrt {5+2{\sqrt {5}}}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 104.5648a^{2}\\V&={\frac {7}{12}}\left(75+37{\sqrt {5}}\right)a^{3}\\&\approx 92.0118a^{3}\end{aligned}}}$
72	Rombicosidodecaedro girado		60	120	62	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&=\left(30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 59.306a^{2}\\V&=\left(20+{\frac {29{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 41.6153a^{3}\end{aligned}}}$
73	Parabigirato rombicosidodecaedro		60	120	62	${\displaystyle D_{5d}}$de orden 20	${\displaystyle {\begin{aligned}A&=\left(30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 59.306a^{2}\\V&=\left(20+{\frac {29{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 41.6153a^{3}\end{aligned}}}$
74	rombicosidodecaedro metabigirato		60	120	62	${\displaystyle C_{2v}}$de orden 4	${\displaystyle {\begin{aligned}A&=\left(30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 59.306a^{2}\\V&=\left(20+{\frac {29{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 41.6153a^{3}\end{aligned}}}$
75	Rombicosidodecaedro trigirado		60	120	62	${\displaystyle C_{3v}}$de orden 6	${\displaystyle {\begin{aligned}A&=\left(30+5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 59.306a^{2}\\V&=\left(20+{\frac {29{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 41.6153a^{3}\end{aligned}}}$
76	Rombicosidodecaedro disminuido		55	105	52	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 58.1147a^{2}\\V&=\left({\frac {115}{6}}+9{\sqrt {5}}\right)a^{3}\\&\approx 39.2913a^{3}\end{aligned}}}$
77	Paragira rombicosidodecaedro disminuido		55	105	52	${\displaystyle C_{5v}}$de orden 10	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 58.1147a^{2}\\V&=\left({\frac {115}{6}}+9{\sqrt {5}}\right)a^{3}\\&\approx 39.2913a^{3}\end{aligned}}}$
78	Rombosidodecaedro disminuido metagirado		55	105	52	${\displaystyle C_{s}}$de orden 2	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 58.1147a^{2}\\V&=\left({\frac {115}{6}}+9{\sqrt {5}}\right)a^{3}\\&\approx 39.2913a^{3}\end{aligned}}}$
79	Bigirato rombicosidodecaedro disminuido		55	105	52	${\displaystyle C_{s}}$de orden 2	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(100+15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 58.1147a^{2}\\V&=\left({\frac {115}{6}}+9{\sqrt {5}}\right)a^{3}\\&\approx 39.2913a^{3}\end{aligned}}}$
80	Rombicosidodecaedro parabidisminuido		50	90	42	${\displaystyle D_{5d}}$de orden 20	${\displaystyle {\begin{aligned}A&={\frac {5}{2}}\left(8+{\sqrt {3}}+2{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 56.9233a^{2}\\V&={\frac {5}{3}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 36.9672a^{3}\end{aligned}}}$
81	Rombicosidodecaedro metabidisminuido		50	90	42	${\displaystyle C_{2v}}$de orden 4	${\displaystyle {\begin{aligned}A&={\frac {5}{2}}\left(8+{\sqrt {3}}+2{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 56.9233a^{2}\\V&={\frac {5}{3}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 36.9672a^{3}\end{aligned}}}$
82	Rombicosidodecaedro bidisminuido girado		50	90	42	${\displaystyle C_{s}}$de orden 2	${\displaystyle {\begin{aligned}A&={\frac {5}{2}}\left(8+{\sqrt {3}}+2{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 56.9233a^{2}\\V&={\frac {5}{3}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 36.9672a^{3}\end{aligned}}}$
83	Rombicosidodecaedro tridisminuido		45	75	32	${\displaystyle C_{3v}}$de orden 6	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(60+5{\sqrt {3}}+30{\sqrt {5+2{\sqrt {5}}}}+9{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 55.732a^{2}\\V&=\left({\frac {35}{2}}+{\frac {23{\sqrt {5}}}{3}}\right)a^{3}\\&\approx 34.6432a^{3}\end{aligned}}}$
84	Disfenoides chato		8	18	12	${\displaystyle D_{2d}}$de orden 8	${\displaystyle {\begin{aligned}A&=3{\sqrt {3}}a^{2}\\&\approx 5.1962a^{2}\\V&\approx 0.8595a^{3}\end{aligned}}}$
85	Antiprisma cuadrado chato		16	40	26	${\displaystyle D_{4d}}$de orden 16	${\displaystyle {\begin{aligned}A&=2\left(1+3{\sqrt {3}}\right)a^{2}\\&\approx 12.3923a^{2}\\V&\approx 3.6012a^{3}\end{aligned}}}$
86	Esfenocorona		10	22	14	${\displaystyle C_{2v}}$de orden 4	${\displaystyle {\begin{aligned}A&=(2+3{\sqrt {3}})a^{2}\\&\approx 7.1962a^{2}\\V&={\frac {1}{2}}a^{3}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}\\&\approx 1.5154a^{3}\end{aligned}}}$
87	Esfenocorona aumentada		11	26	17	${\displaystyle C_{s}}$de orden 2	${\displaystyle {\begin{aligned}A&=(1+4{\sqrt {3}})a^{2}\\&\approx 7.9282a^{2}\\V&={\frac {1}{2}}a^{3}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}+{\frac {1}{3{\sqrt {2}}}}\\&\approx 1.7511a^{3}\end{aligned}}}$
88	Esfenomegacorona		12	28	18	${\displaystyle C_{2v}}$de orden 4	${\displaystyle {\begin{aligned}A&=2\left(1+2{\sqrt {3}}\right)a^{2}\\&\approx 8.9282a^{2}\\V&\approx 1.9481a^{3}\end{aligned}}}$
89	Hebesfenomegacorona		14	33	21	${\displaystyle C_{2v}}$de orden 4	${\displaystyle {\begin{aligned}A&={\frac {3}{2}}\left(2+3{\sqrt {3}}\right)a^{2}\\&\approx 10.7942a^{2}\\V&\approx 2.9129a^{3}\end{aligned}}}$
90	Disfenocingulum		16	38	24	${\displaystyle D_{2d}}$de orden 8	${\displaystyle {\begin{aligned}A&=(4+5{\sqrt {3}})a^{2}\\&\approx 12.6603a^{2}\\V&\approx 3.7776a^{3}\end{aligned}}}$
91	Bilunabirotunda		14	26	14	${\displaystyle D_{2h}}$de orden 8	${\displaystyle {\begin{aligned}A&=\left(2+2{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 12.346a^{2}\\V&={\frac {1}{12}}\left(17+9{\sqrt {5}}\right)a^{3}\\&\approx 3.0937a^{3}\end{aligned}}}$
92	Hebesfenorrotunda triangular		18	36	20	${\displaystyle C_{3v}}$de orden 6	${\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(12+19{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 16.3887a^{2}\\V&=\left({\frac {5}{2}}+{\frac {7{\sqrt {5}}}{6}}\right)a^{3}\\&\approx 5.1087a^{3}\end{aligned}}}$