Outer sphere of polyhedra

Outer sphere of polyhedra
The perfection of the spherical shape has long attracted the attention of thinkers and scientists who, using spheres, tried to explain the harmony of the surrounding world.
 
square inscribed in a circle
Any flat geometric shape can be inscribed inside the circle.
It is rather easy to demonstrate with the example of a square inscribed in a circle. What lies in the word - "described" or "inscribed"?
The definition is as follows: "The circumscribed circle of a polygon is a circle containing all the vertices of the polygon."
That is, for the square, each of the four corners must lie on the circle. In our case, the square is exactly inside the circle and there is no “gap” between the corners and the circle, since all the corners of the square clearly touch the circle. Accordingly, the square was inscribed in a circle. And if we are dealing with three-dimensional geometric bodies, then it can be a matter of inscribing a body in a sphere.
The described sphere is a three-dimensional analogue of the circumcircle, that is, all the vertices of the polyhedron lie on the surface of the sphere. Immediately, we note that not all polyhedra are lucky enough to have their own sphere, but for example, regular polyhedra (tetrahedron, octahedron, cube, dodecahedron, and icosahedron) can boast about it. All regular polyhedra has the described spheres.
Platonic solids inside the sphere
 
 
Escher Order and chaos
Even such a conceptual artist like Maurits Escher did not ignore the connection between the sphere and the polyhedron. In his work "Order and chaos" polyhedron Small stellate dodecahedron is partially inscribed inside the sphere.
 

We managed to find a manufacturer of transparent plastic balls, which can also be disassembled and assembled. And this means the following: almost any of the polyhedra constructed from the “Magic Edges” sets can be placed inside a sphere.
First, it allows us to demonstrate how a sphere describes a polyhedron.
Secondly, it is a very aesthetic design that can be hung, protecting the polyhedron from dust. At the moment we have spheres with the following diameter:
D = 200 mm
D = 180 mm
D = 160 mm
D = 140 mm
D = 120 mm

 

We also have models of Platonic solids from the “Magic Edges №12”, which at first glance may seem of different sizes. But there is one relation that unites all these polytopes with their sizes. This is the diameter of the sphere in which you can put all five models. The dimensions of polyhedra in the 12th issue were chosen so that the sphere described around the polyhedron was exactly 110 mm. This can be especially pronounced when comparing the sizes of the icosahedron and the tetrahedron. And now, having a sphere in place that can be disassembled and assembled, we can successively place there all the Platonic solids, one after the other.

 

There is really one small "but". The manufacturer of these transparent “miracle balls” indicates the diameter of the sphere in outer diameter. And since the sphere has a certain wall thickness and anchoring thickness for the join-disconnect operation, the inner diameter will be slightly smaller.
Creating sets of Platonic and Archimedean bodies, we also adhered to the size of the diameter of the outer sphere.
Why all this? The situation is this. The set of “Magic Edges №12” (Platonic Solids) has the sizes of all five polyhedrons are selected in such a way that they can be inscribed in a sphere with a diameter of 110 mm. And here we are talking about a sphere that is ideally thin from a mathematical point of view. Or about a sphere with an inner diameter of 110 mm. Therefore, if you buy a set of spheres with an outer diameter of 110 mm (there are such ones on sale), then our five Platonic solids simply do not fit there. Why? They simply have an internal diameter of less than 110 mm.
Nevertheless, all our five Platonic solids perfectly fit into balls with an outer diameter of 120 mm. After you close the sphere, there remains a small gap of 4-5 mm, but this can already be attributed to some error.

So, we look - where some polyhedron fits. In a transparent sphere with an outer diameter of 200 mm, polyhedrons can be placed from the issues: 4, 6, 9, 11, 17, 22, and 26.

transparent sphere for models
Small stellated dodecahedron inside the sphere
    Issue № 4
stellated isosahedron inside the sphere
     Issue № 6
Great isosahedron inside the sphere
    Issue № 9
 
 
Compound of five tetrahedra inside the sphere
     Issue № 11
Small stellated dodecahedron inside the sphere
    Issue № 17
Compound three cubes inside the sphere
     Issue № 22
Great dodecicosidodecahedron inside the sphere
     Issue № 26
 
 
 
In a transparent ball with an outer diameter of 180 mm, polyhedrons can be placed from the issues: 5, 8, 10.
transparent sphere for models
Great dodecahedron inside the sphere
    Issue № 5
Great octahedron inside the sphere
     Issue № 8
Small icosicosidodecahedron inside the sphere
     Issue № 10
 
 
In a transparent sphere with an outer diameter of 160 mm, polyhedrons can be placed from the issues: 2, 3, 7, 19, 21.
transparent sphere for models
Dodecadodecahedron inside the sphere
    Issue № 2
Small icosihemidodecahedron inside the sphere
     Issue № 3
small octahedron inside the sphere
 
     Issue № 7
 
Truncated icosahedron inside the sphere
     Issue № 19
icosidodecahedron inside the sphere
         Issue № 19
Small rhombicuboctahedron inside the sphere
    Issue № 21
Truncated cuboctahedron inside the sphere
     Issue № 21
 
 
 
In a transparent sphere with an outer diameter of 140 mm, polyhedrons from issue 18 can be placed.
transparent sphere for models
Truncated tetrahedron inside the sphere
    Issue № 18
Truncated octahedron inside the sphere
    Issue № 18
Truncated cube inside the sphere
     Issue № 18
 
 
Cuboctahedron inside the sphere
     Issue № 18
 
In a transparent sphere with an outer diameter of 120 mm, polyhedrons from the issues can be placed: 12 and 25.
1 120 transparent sphere for models
dodecahedron inside the sphere
    Issue № 12
cube inside the sphere
     Issue № 12
tetrahedron inside the sphere
     Issue № 12
 
 
octahedron inside the sphere
     Issue № 12
isosahedron inside the sphere
    Issue № 12
Cube cross sections inside the sphere
     Issue № 25
Cube cross sections inside the sphere
     Issue № 25
  
 
 
Cube cross sections inside the sphere
     Issue № 25
 
 
A collection of your polyhedra is now under reliable protection!
 
© plyhedr.com  19/02/2018

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