My friend Matt Gorbet showed me this amazing platonic solid video yesterday:

My initial reaction was to wonder if we could build it out of our Zometool set. Luckily we bought some extra green and half green struts last year and were able to figure out how to build it!

My second reaction was to wonder if the icosahedron at the center was the same icosahedron that we’d seen in our “dodecahedron folding into a cube” project:

The last video in the 1st project shows that there is an icosahedron hiding inside of the shape you get when you fold the dodecahdedron:

I wondered if that icosahedron was the same as the one we were seeing in the nested platonic solids. The kids weren’t sure (and neither was I!), so we built the folded shape with the same size dodecahedron at took a closer look:

The last thing we did was connect the orange balls to form the new icosahedron, and – incredibly! – it was exactly the same size as the first one!! How amazing is that – the icosahedron inside of the nested Platonic solids is exactly the same one as you get when you fold up the large dodecahedron.

An excellent observation. I enjoyed playing with the volumes of this order of Platonics. I am a R Buckminster Fuller buff and adhere to his Synergetic Geometry, when expressing volumes in modules and tetrahedral units. The four-fold symmetric forms, cube, tetrahedron and octahedron are easily expressed. From the video, the cube has 24 tetra volumes, the tetrahedron 8 and the octahedron 4.
The icosahedron has a volume of 2.917960 tetra volumes. You can cut and paste the expression to confirm the answer.

(60((phi^-5)/2)+20((phi^-8)/2))

There are 60 S modules and 20 s3 modules or S modules scaled by phi to the next smaller size.

The pentagonal dodecahedron has a volume of 43.416407

912((phi^-5/2)) + 216((phi^-8)/2))

912 S modules and 216 s3 modules.

The volume difference between the cube volume and six “tents” inside the cube when the pentagonal dodecahedron is folded in is 4.583592.

(96((phi^-5)/2)+24((phi^-8)/2))

A tetrahedral unit has an edge of 2 and it s volume in S modules

## Comments

An excellent observation. I enjoyed playing with the volumes of this order of Platonics. I am a R Buckminster Fuller buff and adhere to his Synergetic Geometry, when expressing volumes in modules and tetrahedral units. The four-fold symmetric forms, cube, tetrahedron and octahedron are easily expressed. From the video, the cube has 24 tetra volumes, the tetrahedron 8 and the octahedron 4.

The icosahedron has a volume of 2.917960 tetra volumes. You can cut and paste the expression to confirm the answer.

(60((phi^-5)/2)+20((phi^-8)/2))

There are 60 S modules and 20 s3 modules or S modules scaled by phi to the next smaller size.

The pentagonal dodecahedron has a volume of 43.416407

912((phi^-5/2)) + 216((phi^-8)/2))

912 S modules and 216 s3 modules.

The volume difference between the cube volume and six “tents” inside the cube when the pentagonal dodecahedron is folded in is 4.583592.

(96((phi^-5)/2)+24((phi^-8)/2))

A tetrahedral unit has an edge of 2 and it s volume in S modules

(21((phi^-5)/2)+5((phi^-8)/2)) = 1.000000

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