Higher Dimensional Flower of Life?

[PatrickPowers]

If you are not familiar with the Flower of Life, it can be constructed as follows. Draw a circle. Find the verticies of an inscribed hexagon. For each such vertex draw a circle of that same size with that vertex as center. The pattern can be continued indefinitely with each circle having six circles passing through its center.

All this depends on the length of each side of the inscribed hexagon being equal in length to the radius of the circle. Is there a Platonic solid that would meet this criterion? That is, is there an N-D Platonic solid inscribed within an (N-1)-sphere with radius r such that the distance between neighboring verticies is r? I'm fairly certain this can't be done in arbitrary N, but N=3,4,8,or 24 might be possible.

[Plasmath]

It's not really possible in 3D. Considering that all edge lengths equal 1, none of the circumradii (radius of the circumscribed sphere) of the 3D Platonic solids never equal 1, and only the dodecahedron has a circumradius greater than 1 (it's about 1.4).

However, the cuboctahedron does have a circumradius of 1, but no other Archimedean solids do. So this may be what you're looking for:

cuboctahedron spheres.png

(90.19 KiB) Not downloaded yet

I'm not sure about higher dimensions, however.

[PatrickPowers]

That did the trick. For likely the first time on planet Earth here's the four dimensional Flower of Life. It's made of twelve four-dimensional spheres each with center at a vertex of a cuboctahedron plus another in the center.

https://youtu.be/_jjspK8AtIE

For a 4D arrangement the best you can do is spheres with centers at the verticies of a 24-cell. I'll give that a try tomorrow.

[PatrickPowers]

Here's the 4D Flower of Life with 25 four-dimensional spheres.

https://youtu.be/rKD9J8zj1I4

[Klitzing]

I'd rather advise you to the series of expanded simplices.

x3x (hexagon) is the midsection of x3o3x (cuboctahedron)
x3o3x (cuboctahedron) is the midsection of x3o3o3x (small prismated decachoron)
x3o3o3x (small prismated decachoron) is the midsection of x3o3o3o3x (small cellated dodecateron)
etc.

obviously all have the same circumradius: one edge length.
moreover all those can be thought of as a bistratic lace tegum:

every d+1 dimensional expanded simplex x3o3o...o3o3x (d+1 nodes) consists out of 3 vertex layers,
which represent a regular d-dimensional simplex x3o3o...o3o (d nodes) at the top,
the d-dimensional expanded simplex x3o3o...o3x (d nodes) in the middle,
and the d-dimensional regular dual simplex o3o3o...o3x (d nodes) at the bottom.

--- rk

[PatrickPowers]

Here are two simpler versions.

The usual 2D hexagonal Flower of Life made of circles rotating in four dimensional space. https://youtu.be/YknFIBmDFHw

The usual 2D hexagonal Flower of Life but made of 3-spheres instead of 1-spheres and rotating in four dimensional space. https://youtu.be/_9dAp3J7gJM