Let be the inradius of the dual polyhedron corresponding to the insphere , which touches the faces of the dual solid , be the midradius of both the polyhedron and its dual corresponding to the midsphere , which touches the edges of both the polyhedron and its duals , the circumradius corresponding to the circumsphere of the solid which touches the vertices of the solid of the Platonic solid, and the edge length of the solid.
Since the circumsphere and insphere are dual to each other, they obey the relationship. The following two tables give the analytic and numerical values of these distances for Platonic solids with unit side length. Finally, let be the area of a single face , be the volume of the solid, and the polyhedron edges be of unit length on a side.
The following table summarizes these quantities for the Platonic solids. The following table gives the dihedral angles and angles subtended by an edge from the center for the Platonic solids Cundy and Rollett , Table II following p. The number of polyhedron edges meeting at a polyhedron vertex is. For the solid whose faces are -gons denoted , with touching at each polyhedron vertex , the symbol is. Given and , the number of polyhedron vertices , polyhedron edges , and faces are given by.
The plots above show scaled duals of the Platonic solid embedded in an augmented form of the original solid, where the scaling is chosen so that the dual vertices lie at the incenters of the original faces Wenninger , pp. Since the Platonic solids are convex, the convex hull of each Platonic solid is the solid itself.
Minimal surfaces for Platonic solid frames are illustrated in Isenberg , pp. Artmann, B. Washington, DC: Math. Atiyah, M. Ball, W. New York: Dover, pp. Behnke, H. Fundamentals of Mathematics, Vol. Beyer, W. Bogomolny, A. Bourke, P. Coxeter, H. Regular Polytopes, 3rd ed. Critchlow, K. New York: Viking Press, Cromwell, P. New York: Cambridge University Press, pp.
Cundy, H. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub. Dunham, W. New York: Wiley, pp. New York: Dover, Gardner, M. New York: Simon and Schuster, pp. Geometry Technologies. A tetrahedron is known as a triangular pyramid in geometry.
The tetrahedron consists of 4 triangular faces, 6 straight edges, and 4 vertex corners. It is a platonic solid which has a three-dimensional shape with all faces as triangles. The properties of a tetrahedron are:. A cube is a 3D solid object with 6 square faces and all the sides of a cube are of the same length. The cube is also known as a regular hexahedron that is a box-shaped solid with 6 identical square faces. The properties of a cube are:.
An octahedron is a polyhedron with 8 faces, 12 edges, and 6 vertices and at each vertex 4 edges meet. The faces of an octahedron are shaped like an equilateral triangle. The properties of an octahedron are:. A dodecahedron is a platonic solid that consists of 12 sides and 12 pentagonal faces. The properties of a dodecahedron are:. The properties of an icosahedron are:. Platonic solids are considered to be only 5 solid shapes. Here are the reasons why there are only 5 shapes and not more:.
According to Euler's formula, for any convex polyhedron, the Number of Faces plus the Number of Vertices corner points minus the Number of Edges always equals 2.
Let us take apply this in one of the platonic solids - Icosahedron. Example 1: Demi wants to know the name of the platonic solid shown below.
Can you name the platonic solid for her? Solution: The given solid has 20 triangular faces and 5 triangles are intersecting at each vertex, which is a property of an icosahedron. Hence the given solid is an icosahedron. Example 2: Rita was given the following information about a platonic solid that it has 3 faces meeting at vertices and has 4 vertices.
The tetrahedron is self-dual i. The cube and the octahedron form a dual pair. The dodecahedron and the icosahedron form a dual pair.
There cannot be a platonic solid made up of hexagons — even if three hexagons meet at a vertex this will create an angle of which is too big. Any others are not possible because the internal angles are too big. This means that the strengths of 2D and 3D shapes are independent of one another. The project also ended with a conclusion that the cube, tetrahedron, and octahedron are the strongest Platonic solids.
The world as we know it has three dimensions of space —length, width and depth—and one dimension of time. But there's the mind-bending possibility that many more dimensions exist out there. According to string theory, one of the leading physics model of the last half century, the universe operates with 10 dimensions.
A prism is a solid structure with flat faces and identical faces at both ends. As a result, all prisms are NOT platonic solids. There have only been 5 platonic solids: the tetrahedron, the octahedron, the icosahedron, the cube, and the dodecahedron. Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental.
Platonism in this sense is a contemporary view. Of all Platonic solids only the tetrahedron, cube, and octahedron occur naturally in crystal structures. The regular icosahedron and dodecahedron are not amongst the crystal habit.
The icosahedron — sided polyhedron — is frequent. An Archimedean solid is a polyhedron made up of different kinds of regular polygons, that looks the same from every direction. There are 13 different Archimedean solids. A regular polygon is a polygon in which all sides have the same length and all interior angles have the same size.
Altogether there are only five regular solids. The remaining three are the octahedron, the dodecahedron, and the icosahedron. The fact that there are only five regular solids can be traced to Euclid, who devotes much of the final chapter of his work the Elements to various facts about the regular solids. Why are platonic solids called platonic solids?
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