Natural Bronze
Nested Platonic Solids (Version Sd)
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Product Description
This nesting of the 5 platonic solids underlines the symmetries and the dualities of their geometries.
The tetrahedron is the main element of the structure.
The two other triangular-face platonic solids (octahedron and icosahedron) are nested inside it with shared faces and common inscribed sphere (of radius 1 cm).
The icosahedron vertices intersect the octahedron edges following the golden ratio phi (1.618...).
The tetrahedron is nested inside the non-triangular-face platonic solids (hexahedron and dodecahedron) with shared vertices and common circumscribed sphere (of radius 3 cm).
The exact ratio of 3 between the two spheres can be checked with the tetrahedron formula since it is the only solid in this nesting related to the two spheres (https://en.wikipedia.org/wiki/Tetrahedron).
The same nesting can be infinitely repeated with solids 3 times smaller (edge wise) at each step.
The duality of hexahedron/octahedron and dodecahedron/icosahedron is shown since vertices are aligned with faces.
The inscribed sphere is made of 6 great circles following the geometry of the tetrakis hexahedron (a Catalan solid, intermediate between a cube and a rhombic dodecahedron). The eight points where 3 great circles crosses are aligned with the octahedron faces and cube vertices. It is at these points that the sphere is tangent with the octahedron faces.
These 8 faces are the one used in the nice Jitterbug animation showing the transformation between the octahedron and the cuboctahedron (https://www.antiprism.com/album/misc/jit_inout.gif).
The icosahedron appears between these two shapes (golden ratio).
The circumscribed sphere is made of 3 great circles and represent 8 equilateral-rectangular spherical triangles.
This nesting is availalbe in 5 different versions :
- Version S : A sphere is at the center (Sphere-to-Sphere Nesting)
- Version Sd :Same as S with 'duality' lines connecting 8 dodecahedron vertices to the corresponding faces on the icosahedron where 3 great circles of the inscrebed sphere meet.
- Version D : A dodecahedron is at the center, the inscribed sphere is represent by 3 great circles (same as the circumscribed sphere)
- Version Dd : Same as D with 'duality' lines connecting 8 dodecahedron vertices to the corresponding faces on the icosahedron and to the smaller dodecahedron faces. The spheres are not shown (Dodecahedron-to-Dodecahedron Nesting)
- Version T : A smaller tetrahedron is at the center. This second tetrahedron is the dual of the first one (Tetrahedron-to-Tetrahedron Nesting)
The wire diameter is 1 mm (total diameter of 60 + 1 mm).
I would like to thank Rick Russel who encouraged me to produce this model for the Marketplace.
The tetrahedron is the main element of the structure.
The two other triangular-face platonic solids (octahedron and icosahedron) are nested inside it with shared faces and common inscribed sphere (of radius 1 cm).
The icosahedron vertices intersect the octahedron edges following the golden ratio phi (1.618...).
The tetrahedron is nested inside the non-triangular-face platonic solids (hexahedron and dodecahedron) with shared vertices and common circumscribed sphere (of radius 3 cm).
The exact ratio of 3 between the two spheres can be checked with the tetrahedron formula since it is the only solid in this nesting related to the two spheres (https://en.wikipedia.org/wiki/Tetrahedron).
The same nesting can be infinitely repeated with solids 3 times smaller (edge wise) at each step.
The duality of hexahedron/octahedron and dodecahedron/icosahedron is shown since vertices are aligned with faces.
The inscribed sphere is made of 6 great circles following the geometry of the tetrakis hexahedron (a Catalan solid, intermediate between a cube and a rhombic dodecahedron). The eight points where 3 great circles crosses are aligned with the octahedron faces and cube vertices. It is at these points that the sphere is tangent with the octahedron faces.
These 8 faces are the one used in the nice Jitterbug animation showing the transformation between the octahedron and the cuboctahedron (https://www.antiprism.com/album/misc/jit_inout.gif).
The icosahedron appears between these two shapes (golden ratio).
The circumscribed sphere is made of 3 great circles and represent 8 equilateral-rectangular spherical triangles.
This nesting is availalbe in 5 different versions :
- Version S : A sphere is at the center (Sphere-to-Sphere Nesting)
- Version Sd :Same as S with 'duality' lines connecting 8 dodecahedron vertices to the corresponding faces on the icosahedron where 3 great circles of the inscrebed sphere meet.
- Version D : A dodecahedron is at the center, the inscribed sphere is represent by 3 great circles (same as the circumscribed sphere)
- Version Dd : Same as D with 'duality' lines connecting 8 dodecahedron vertices to the corresponding faces on the icosahedron and to the smaller dodecahedron faces. The spheres are not shown (Dodecahedron-to-Dodecahedron Nesting)
- Version T : A smaller tetrahedron is at the center. This second tetrahedron is the dual of the first one (Tetrahedron-to-Tetrahedron Nesting)
The wire diameter is 1 mm (total diameter of 60 + 1 mm).
I would like to thank Rick Russel who encouraged me to produce this model for the Marketplace.
Details
What's in the box:
Nesting.Sd
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Rating:
Mature audiences only.
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