# inside the cube

The sequences of images in this webpage shows how the convex regular polyhedra (aka platonic solids) and polyhedron compounds (of the convex regular) are obtained from the cube:

### Regular facets of the cube

From the vertices of the cube, it is only possible to obtain one type of regular facet: the regular triangle. With four of these, we obtain the regular tetrahedron, as the following will show.

### Tetrahedra inside the cube

The face diagonals of the cube are the edges of the tetrahedron, and two different tetrahedra are thus obtainable.

The union of two equal tetrahedra with the same centroid is the polyhedron compound known as stella octangula; their intersection is the regular octahedron. The stella octangula is not a polyhedron, but a compound, because four of its vertices derive from the intersection of two faces (instead of, at least, three). The stella octangula is a regular compound [2: 47-48] because it shares its vertices and faces with two other regular polyhedra: the cube and the octahedron, respectively.

### Octahedron inside the cube 01

The cube and the octahedron are dual to each other, so the face centroids of the cube are the vertices of the octahedron. In this situation, however, they do not share the same reciprocation sphere.

### Octahedra inside the cube 02

The octahedron may also be inscribed in the cube by dividing its edges into 4 equal parts and suitably connecting the points at 3/4 of each edge. The edges of the octahedron are thus parallel to the face diagonals of the cube. three more octahedra are inscribable within the cube, from which the compound of four octahedra is obtained.

### Dodecahedra inside the cube 01

• divide one of the edges of the cube in the golden ratio (in the image below, we chose to do so in the upper face of the cube, by interior division of one of its edges)
• the shorter segment has the length of the dodecahedron’s edge, so we translate this to a median of the upper face
• by transposing this segment to the remaining faces, we obtain 12 vertices of the dodecahedron
• the segment parallel to two faces of the cube that connects the midpoint of the first edge to a specific vertex is a diagonal of one of the pentagons. for the remaining vertices of this pentagon, we consider the plane that contains this diagonal and is perpendicular to those faces of the cube
• we repeat this procedure to another diagonal and another one still
• the intersection of three of the three planes gives us another vertex of the dodecahedron
• by repeating the procedure or through rotational symmetry, we obtain the remaining vertices of the dodecahedron

a second dodecahedron is similarly inscribable within the cube. its edges are perpendicular to the initial dodecahedron:

### Dodecahedron from the cube 02

this is an exploration of Euclid’s XIII.17 and Kepler’s construction based upon it , from which a dodecahedron is obtained from the cube.

• we take one of the square’s apothems and divide it in the golden ratio, by its interior division
• from this division, we draw a perpendicular to the cube’s faces and transpose the larger segment of the golden ratio to it with a sphere or a circle. the point we have just determined is one of the dodecahedron’s vertices
• we take the centroid of the adjacent face to draw a sphere or circle with the same radius as the larger segment of the golden ratio
• from its intersection with two apothems, two segments are drawn, perpendicular to the face of the cube, with the same length as the first perpendicular that gave us one vertex of the dodecahedron
• the points we have determined and the endpoints of the edge common to the two faces of the cub will allows us to obtain a face of the dodecahedron
• a similar procedure in an adjacent face allows us to obtain another face
• by repeating the procedure or through rotational symmetry, we obtain the remaining vertices of the dodecahedron

### Icosahedra inside the cube

• divide one of the edges of the cube in the golden ratio (in the image below, we chose to do so in the upper face of the cube, by interior division of one of its edges)
• the longest segment has the length of the icosahedron’s edge, so we translate this to a median of the upper face of the cube
• by transposing this segment to the remaining faces, we obtain the 12 vertices of the icosahedron
• these are also the vertices of three mutually perpendicular golden rectangles

:

a second icosahedron is similarly inscribable within the cube. its edges are perpendicular to the initial icosahedron.

References:

1. Pugh, A. (1976). Polyhedra: A visual approach. Berkeley: University of California Press.
2. Coxeter, H. (1973). Regular Polytopes. New York: Dover Publications.
3. Hart, G. (1998) Johannes Kepler’s Polyhedra. https://www.georgehart.com/virtual-polyhedra/kepler.html
4. Viana, V. 2020. Aplicações didácticas sobre poliedros para o ensino da geometria / Didactic Applications on Polyhedra for the teaching of Geometry (http://hdl.handle.net/10348/10337) [Tese de Doutoramento, Universidade de Trás-os-Montes e Alto Douro]. Repositório Científico da UTAD.