File:Icosaedro Estrellado Dr, Leonel Fernández.jpg

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It is a three-dimensional polytope that is structured by 60 uniform faces, which have the shape of Jose's isosceles triangles, has 32 non-uniform vertices and 90 non-uniform edges.

Summary

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Description
Español: Icosaedro Estrellado Leonel Fernández: Es un politopo tridimensional que está estructurado por 60 caras que poseen forma de triángulos isósceles de Jose (C = caras), posee 12 vértices intermedios (V = Vértices), 20 vértices exteriores para un total de 32 vértices, además tiene 60 aristas exteriores y 30 aristas intermedia para un total de 90 aristas (A = aristas). Si aplicamos la fórmula de poliedro de Leonard Euler A = C+V-2. Datos: A = 90, V = 32, C = 60. Sustituyendo en la fórmula de Euler tenemos: (A = C+V-2) = ((90) = (60) + (32) -2) = (90 = 60 +32-2) = 90 = 90, comprobamos que se verifica la igualdad. Con los puntos que están ubicados en cada uno de los centros de las caras del icosidodecaedro de Arquímedes, se define totalmente el icosaedro estrellado Leonel Fernández, por lo tanto son poliedros duales entre sí.

Estos puntos definen el icosaedro estrellado Leonel Fernández el cual se refleja en la parte interior del solido conocido con el nombre del icosidodecaedro de Arquímedes. Observemos: A = (0.688190960235586, -0.5, 1.376381920471174), B = (-0.262865556059567, -0.809016994374947, 1.376381920471174), C = (-0.85065080835204, 0, 1.376381920471174), D = (0.688190960235586, 0.5 ,1.376381920471174), E = (-0.262865556059567, 0.809016994374947 ,1.376381920471174), F = (-1.113516364411607, 0.809016994374948, 0.85065080835204), G = (0.425325404176019, 1.309016994374948, 0.85065080835204), H = (1.376381920471173, 0, 0.85065080835204), I = (0.425325404176019, -1.309016994374948, 0.85065080835204), J = (-1.113516364411607,-0.809016994374948, 0.85065080835204), K = (-1.538841768587627, 0.5, 0), L = (-0.951056516295152, 1.309016994374948, 0), M = (0.000000000000001, 1.618033898749895, 0), N = (0.951056516295154, 1.309016994374947, 0), O = (1.538841768587627, 0.5, 0), P = (1.538841768587627, -0.5, 0), Q = (0.951056516295154, -1.309016994374948, 0), R = (0, -1.618033898749895, 0), S = (-0.951056516295153, -1.309016994374947, 0), T = (-1.538841768587627, -0.5, 0), U = (-0.425325404176019, 1.309016994374947, -0.85065080835204), V = (1.113516364411607, 0.809016994374947, -0.85065080835204), W = (1.113516364411607, -0.809016994374948, -0.85065080835204), Z = (-0.425325404176019, -1.309016994374947, -0.85065080835204), A1 = (-1.376381920471173, 0, -0.85065080835204), B1 = (0.262865556059567, 0.809016994374947, -1.376381920471174), C1 = (0.85065080835204, 0, -1.376381920471174), D1 = (0.262865556059567, -0.809016994374948, -1.376381920471174), E1 = (-0.688190960235586, -0.5, -1.376381920471174), F1 = (-0.688190960235586, 0.5, -1.376381920471174), G1 = (0, 0, 1.376381920471174), H1 = (0.283550269450679, -0.872677996249965, 1.201138216431463), I1 = (-0.742344242941071, -0.539344662916632, 1.201138216431463), J1 = (-0.742344242941071, 0.539344662916632, 1.201138216431463), K1 = (0.283550269450679, 0.872677996249965, 1.201138216431463, L1 = (0.917587946980782, 0, 1.201138216431463), M1 = (-0.380422606518061, -1.170820393249937, 0.615536707435051), N1 = (-1.231073414870102, 0, 0.615536707435051), O1 = (-0.380422606518061, 1.170820393249937, 0.615536707435051), P1 = (0.995959313953112, 0.723606797749979, 0.615536707435051), Q1 = (0.995959313953112, -0.723606797749979, 0.615536707435051), R1 = (-1.201138216431462, -0.872677996249965, 0.28355026945068), S1 = (-1.201138216431462, 0.872677996249965, 0.28355026945068), T1 = (0.458793973490391, 1.412022659166597, 0.28355026945068), U1 = (1.484688485882142, 0, 0.28355026945068), V1 = (0.458793973490391, -1.412022659166597, 0.28355026945068), W1 = (-0.458793973490391, -1.412022659166596, -0.28355026945068), Z1 = (-1.484688485882142, 0, -0.28355026945068), A2 = (-0.458793973490390, 1.412022659166596, -0.28355026945068), B2 = (1.201138216431463, 0.872677996249965, -0.28355026945068), C2 = (1.201138216431463, -0.872677996249965, -0.28355026945068), D2 = (-0.995959313953112, -0.723606797749979, -0.615536707435051), E2 = (-0.995959313953112, 0.723606797749979, -0.615536707435051), F2 = (0.380422606518061, 1.170820393249937, -0.615536707435051), G2 = (1.231073414870102, 0, -0.615536707435051), H2 = (0.38042260651806156, -1.170820393249937, -0.615536707435051), I2 = (-0.283550269450679, -0.872677996249965, -1.201138216431463), J2 = (-0.917587946980782, 0, -1.201138216431463), K2 = (-0.283550269450679, 0.872677996249965, -1.201138216431463), L2 = (0.742344242941071, 0.539344662916631, -1.201138216431463), M2 = (0.742344242941071, -0.539344662916632, -1.201138216431463), N2 = (0, 0, -1.376381920471174).

Fue descubierto por el Dominicano Jose Joel Leonardo el 5 de enero del 2020 a las 10:20 Pm y nombrado por su autor el 26 de Diciembre del 2022, en honor escritor y político dominicano Dr. Leonel Antonio Fernández Reina.
English: isosceles triangles (C = faces), it has 12 intermediate vertices (V = Vertices), 20 exterior vertices for a total of 32 vertices, in addition it has 60 exterior edges and 30 intermediate edges for a total of 90 edges (A = edges). If we apply the Leonard Euler polyhedron formula A = C+V-2. Data: A = 90, V = 32, C = 60. Substituting into Euler's formula we have: (A = C+V-2) = ((90) = (60) + (32) -2) = (90 = 60 +32-2) = 90 = 90, we check that equality is verified. With the points that are located in each of the centers of the faces of the Archimedean icosidodecahedron, the Leonel Fernández stellated icosahedron is fully defined, therefore they are dual polyhedra among themselves.

These points define the stellated Leonel Fernández icosahedron, which is reflected in the inner part of the solid known as the Archimedean icosidodecahedron. Let's observe: A = (0.688190960235586, -0.5, 1.376381920471174), B = (-0.262865556059567, -0.809016994374947, 1.376381920471174), C = (-0.85065080835204, 0, 1.376381920471174), D = (0.688190960235586, 0.5 ,1.376381920471174), E = (-0.262865556059567, 0.809016994374947 ,1.376381920471174), F = (-1.113516364411607, 0.809016994374948, 0.85065080835204), G = (0.425325404176019, 1.309016994374948, 0.85065080835204), H = (1.376381920471173, 0, 0.85065080835204), I = (0.425325404176019, -1.309016994374948, 0.85065080835204), J = (-1.113516364411607,-0.809016994374948, 0.85065080835204), K = (-1.538841768587627, 0.5, 0), L = (-0.951056516295152, 1.309016994374948, 0), M = (0.000000000000001, 1.618033898749895, 0), N = (0.951056516295154, 1.309016994374947, 0), O = (1.538841768587627, 0.5, 0), P = (1.538841768587627, -0.5, 0), Q = (0.951056516295154, -1.309016994374948, 0), R = (0, -1.618033898749895, 0), S = (-0.951056516295153, -1.309016994374947, 0), T = (-1.538841768587627, -0.5, 0), U = (-0.425325404176019, 1.309016994374947, -0.85065080835204), V = (1.113516364411607, 0.809016994374947, -0.85065080835204), W = (1.113516364411607, -0.809016994374948, -0.85065080835204), Z = (-0.425325404176019, -1.309016994374947, -0.85065080835204), A1 = (-1.376381920471173, 0, -0.85065080835204), B1 = (0.262865556059567, 0.809016994374947, -1.376381920471174), C1 = (0.85065080835204, 0, -1.376381920471174), D1 = (0.262865556059567, -0.809016994374948, -1.376381920471174), E1 = (-0.688190960235586, -0.5, -1.376381920471174), F1 = (-0.688190960235586, 0.5, -1.376381920471174), G1 = (0, 0, 1.376381920471174), H1 = (0.283550269450679, -0.872677996249965, 1.201138216431463), I1 = (-0.742344242941071, -0.539344662916632, 1.201138216431463), J1 = (-0.742344242941071, 0.539344662916632, 1.201138216431463), K1 = (0.283550269450679, 0.872677996249965, 1.201138216431463, L1 = (0.917587946980782, 0, 1.201138216431463), M1 = (-0.380422606518061, -1.170820393249937, 0.615536707435051), N1 = (-1.231073414870102, 0, 0.615536707435051), O1 = (-0.380422606518061, 1.170820393249937, 0.615536707435051), P1 = (0.995959313953112, 0.723606797749979, 0.615536707435051), Q1 = (0.995959313953112, -0.723606797749979, 0.615536707435051), R1 = (-1.201138216431462, -0.872677996249965, 0.28355026945068), S1 = (-1.201138216431462, 0.872677996249965, 0.28355026945068), T1 = (0.458793973490391, 1.412022659166597, 0.28355026945068), U1 = (1.484688485882142, 0, 0.28355026945068), V1 = (0.458793973490391, -1.412022659166597, 0.28355026945068), W1 = (-0.458793973490391, -1.412022659166596, -0.28355026945068), Z1 = (-1.484688485882142, 0, -0.28355026945068), A2 = (-0.458793973490390, 1.412022659166596, -0.28355026945068), B2 = (1.201138216431463, 0.872677996249965, -0.28355026945068), C2 = (1.201138216431463, -0.872677996249965, -0.28355026945068), D2 = (-0.995959313953112, -0.723606797749979, -0.615536707435051), E2 = (-0.995959313953112, 0.723606797749979, -0.615536707435051), F2 = (0.380422606518061, 1.170820393249937, -0.615536707435051), G2 = (1.231073414870102, 0, -0.615536707435051), H2 = (0.38042260651806156, -1.170820393249937, -0.615536707435051), I2 = (-0.283550269450679, -0.872677996249965, -1.201138216431463), J2 = (-0.917587946980782, 0, -1.201138216431463), K2 = (-0.283550269450679, 0.872677996249965, -1.201138216431463), L2 = (0.742344242941071, 0.539344662916631, -1.201138216431463), M2 = (0.742344242941071, -0.539344662916632, -1.201138216431463), N2 = (0, 0, -1.376381920471174).

It was discovered by the Dominican Jose Joel Leonardo on January 5, 2020 at 10:20 pm and named by its author on December 26, 2022, in honor of the Dominican writer and politician Dr. Leonel Antonio Fernández Reina.
Français : Leonel Fernández Starry Icosahedron : C'est un polytope tridimensionnel structuré par 60 faces qui ont la forme des triangles isocèles de Jose (C = faces), il a 12 sommets intermédiaires (V = Vertices), 20 sommets extérieurs pour un total de 32 sommets, en plus il a 60 arêtes extérieures et 30 arêtes intermédiaires pour un total de 90 arêtes (A = arêtes). Si nous appliquons la formule du polyèdre de Leonard Euler A = C+V-2. Données : A = 90, V = 32, C = 60. En remplaçant dans la formule d'Euler, nous avons : (A = C+V-2) = ((90) = (60) + (32) -2) = (90 = 60 +32-2) = 90 = 90, on vérifie que l'égalité est vérifiée. Avec les points situés dans chacun des centres des faces de l'icosidodécaèdre d'Archimède, l'icosaèdre étoilé de Leonel Fernández est entièrement défini, ce sont donc des polyèdres doubles entre eux.

Ces points définissent l'icosaèdre étoilé de Leonel Fernández, qui se reflète dans la partie interne du solide connue sous le nom d'icosidodécaèdre d'Archimède. Observons : A = (0.688190960235586, -0.5, 1.376381920471174), B = (-0.262865556059567, -0.809016994374947, 1.376381920471174), C = (-0.85065080835204, 0, 1.376381920471174), D = (0.688190960235586, 0.5 ,1.376381920471174), E = (-0.262865556059567, 0.809016994374947 ,1.376381920471174), F = (-1.113516364411607, 0.809016994374948, 0.85065080835204), G = (0.425325404176019, 1.309016994374948, 0.85065080835204), H = (1.376381920471173, 0, 0.85065080835204), I = (0.425325404176019, -1.309016994374948, 0.85065080835204), J = (-1.113516364411607,-0.809016994374948, 0.85065080835204), K = (-1.538841768587627, 0.5, 0), L = (-0.951056516295152, 1.309016994374948, 0), M = (0.000000000000001, 1.618033898749895, 0), N = (0.951056516295154, 1.309016994374947, 0), O = (1.538841768587627, 0.5, 0), P = (1.538841768587627, -0.5, 0), Q = (0.951056516295154, -1.309016994374948, 0), R = (0, -1.618033898749895, 0), S = (-0.951056516295153, -1.309016994374947, 0), T = (-1.538841768587627, -0.5, 0), U = (-0.425325404176019, 1.309016994374947, -0.85065080835204), V = (1.113516364411607, 0.809016994374947, -0.85065080835204), W = (1.113516364411607, -0.809016994374948, -0.85065080835204), Z = (-0.425325404176019, -1.309016994374947, -0.85065080835204), A1 = (-1.376381920471173, 0, -0.85065080835204), B1 = (0.262865556059567, 0.809016994374947, -1.376381920471174), C1 = (0.85065080835204, 0, -1.376381920471174), D1 = (0.262865556059567, -0.809016994374948, -1.376381920471174), E1 = (-0.688190960235586, -0.5, -1.376381920471174), F1 = (-0.688190960235586, 0.5, -1.376381920471174), G1 = (0, 0, 1.376381920471174), H1 = (0.283550269450679, -0.872677996249965, 1.201138216431463), I1 = (-0.742344242941071, -0.539344662916632, 1.201138216431463), J1 = (-0.742344242941071, 0.539344662916632, 1.201138216431463), K1 = (0.283550269450679, 0.872677996249965, 1.201138216431463, L1 = (0.917587946980782, 0, 1.201138216431463), M1 = (-0.380422606518061, -1.170820393249937, 0.615536707435051), N1 = (-1.231073414870102, 0, 0.615536707435051), O1 = (-0.380422606518061, 1.170820393249937, 0.615536707435051), P1 = (0.995959313953112, 0.723606797749979, 0.615536707435051), Q1 = (0.995959313953112, -0.723606797749979, 0.615536707435051), R1 = (-1.201138216431462, -0.872677996249965, 0.28355026945068), S1 = (-1.201138216431462, 0.872677996249965, 0.28355026945068), T1 = (0.458793973490391, 1.412022659166597, 0.28355026945068), U1 = (1.484688485882142, 0, 0.28355026945068), V1 = (0.458793973490391, -1.412022659166597, 0.28355026945068), W1 = (-0.458793973490391, -1.412022659166596, -0.28355026945068), Z1 = (-1.484688485882142, 0, -0.28355026945068), A2 = (-0.458793973490390, 1.412022659166596, -0.28355026945068), B2 = (1.201138216431463, 0.872677996249965, -0.28355026945068), C2 = (1.201138216431463, -0.872677996249965, -0.28355026945068), D2 = (-0.995959313953112, -0.723606797749979, -0.615536707435051), E2 = (-0.995959313953112, 0.723606797749979, -0.615536707435051), F2 = (0.380422606518061, 1.170820393249937, -0.615536707435051), G2 = (1.231073414870102, 0, -0.615536707435051), H2 = (0.38042260651806156, -1.170820393249937, -0.615536707435051), I2 = (-0.283550269450679, -0.872677996249965, -1.201138216431463), J2 = (-0.917587946980782, 0, -1.201138216431463), K2 = (-0.283550269450679, 0.872677996249965, -1.201138216431463), L2 = (0.742344242941071, 0.539344662916631, -1.201138216431463), M2 = (0.742344242941071, -0.539344662916632, -1.201138216431463), N2 = (0, 0, -1.376381920471174).

Il a été découvert par le dominicain José Joel Leonardo le 5 janvier 2020 à 22h20 et nommé par son auteur le 26 décembre 2022, en l'honneur de l'écrivain et homme politique dominicain Dr Leonel Antonio Fernández Reina.
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Author Jose J. Leonard

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