Icosahedron turned to sphere, just for fun

[Robert Marshall]

The icosahedron ("icosa" is Greek for 20) is one of 5 Platonic solids (3-dimensional geometric shapes that are comprised of identical polygons). It is made up of 20 equilateral triangles. (The other Platonic forms are the tetrahedron (4 equilateral triangles), the hexahedron (aka, "cube"; 6 squares), the octohedron (8 equilateral triangles), and the dodecahedron (12 pentagons)).

The steps to build an icosahedron are these:

1. Cut 20 equilateral triangles of the same size. I used a compound miter saw to do this, cutting a narrow pine board. Set angle at 30 degrees, cut, flip, cut, flip, etc. Two 30 degree cuts give you 60 degree triangles. Cut a few extra in case you screw something up.
2. Bevel all 3 edges of 20 triangles, to 20.9 degrees off vertical. I built a simple table-saw sled with an equilateral notch and a hold-down clamp, photo below. Used a digital angle measurement tool, to get the saw blade set to 20.9 degrees.
3. Arrange the beveled triangles, bevels down, into the depicted arrangement, diagram below. (There are apparently 43,880 ways to arrange the triangles and still get the same icosahedron shape, but this seemed simple enough for me.)
4. Tape them to each other. I used gaffers tape, from Amazon; holds well, leaves no residue.
5. Flip the taped arrangement over. Probably best to arrange on a small sheet of plywood, and place another sheet over the arrangement, for flipping, because it can be a floppy mess, otherwise. This exposes the joints which you will soon glue-up.
6. Carefully do a "dry-assembly," folding the flat arrangement into its intended shape, without gluing yet. (If it won't fold into an icosahedron, you've got something wrong with your triangles; maybe not identical size, maybe not 20.9 degrees bevel.) Then unfold it carefully, so that the joints that are to be glued are face-up.
7. Apply glue to all the beveled faces. I used Titebond Quick and Thick; its thickness avoided lots of runny mess and helped fill any gaps, and its quickness made it, well ... quick. Fold the arrangement into an icosahedron. Requires some pushing and shoving to get everything into its proper place; don't waste any time at this stage, because the glue is, as promised, "quick." Photo of finished icosahedron, below.

Once I was confident it could handle being turned, I put it between centers, using the midpoints of two opposite triangles. Every triangle has an opposite side, pick any two. Then I turned it into a sphere, except for the two triangles where the centers were placed. I then took off the lathe, and rotated it 90 degrees, and placed it back on the lathe, this time with the centers placed at two opposite vertices (where 5 triangles meet). This enabled me to get the portion where the centers were originally placed trued-up to spherical shape (or close enough, anyway). I used a sharp negative rake rounded-nose scraper, and sandpaper, to do the rounding. Photo of sphere below,

Beveling sled:
Screenshot 2019-12-08 at 9.41.01 AM.jpg

Arrangement of triangles for assembly:
icosahedron.png

Finished icosahedron, before turning:
Screenshot 2019-12-08 at 9.26.18 AM.jpg

Sphere, turned from icosahedron:
Screenshot 2019-12-08 at 9.29.01 AM.png

Inspired by Malcolm Tibbett's book on segmented woodturning, and also Barry McFadden's "Plywood Ball" post (https://sawmillcreek.org/showthread....5-Plywood-Ball). (I'm still not clear what shape Barry's son built and turned, except that it was NOT an icosahedron.)

[John K Jordan]

....
Inspired by Malcolm Tibbett's book on segmented woodturning, and also Barry McFadden's "Plywood Ball" post (https://sawmillcreek.org/showthread....5-Plywood-Ball). (I'm still not clear what shape Barry's son built and turned, except that it was NOT an icosahedron.)

Very nice! Clever design and jig. Looks like a bunch of work!

Looking at the vertices of the one that Barry's son made, I'm imagining it might be convex regular icosahedron like you made except where each of the 20 segments was made of four smaller triangles glued together flat, all with 90-deg joints at the three interior edges to make the 20 flat triangles. The outer three edges would have to be beveled as you showed. This would explain the arrangement at the vertices visible in his photos, five where the composite triangles meet and six at what would be the midpoints of each side of your 20 triangles. Hard to know for sure without holding it in the hand.

Did you see the video of the truncated icosahedron sphere (soccer ball) that was in Barry's post? https://boingboing.net/2018/11/21/ho...-geodesic.html

Regardless, I'm impressed with both yours and Barry's son's spheres (and the one in the video). But I think I'll stick to turning solid wood!

BTW, I use a lot of gaffers tape. (the real stuff, not the cheap imitation) Another tape that works well for lots of shop and home tasks is this 3M/Scotch #2060: https://www.amazon.com/gp/product/B00004Z4AY (This is not the green tape found at the big box stores)

JKJ

[tom lucas]

fun little project. Nicely done. Thanks for sharing.

[Robert Marshall]

Appreciate the comments.

I wonder if the "plywood ball" that Barry McFadden posted about was a Pentakis dodecahedron. It is the shape you get when you put a 5 triangle pyramid on each of the 12 pentagonal faces of a dodecahedron, so it has 60 (5 * 12) faces, each face an isosceles triangle Here is a description/calculator for that shape:

https://rechneronline.de/pi/pentakis-dodecahedron.php

I think that might be it, because it gives you a combination of pentagon shapes (the pyramids), and also hexagon shapes (each hexagon shape formed from 2 adjacent triangles in one pyramid, 2 triangles in one of the abutting pyramids, and 2 triangles from the other abutting pyramid.) The pyramids are "intentional" (built) shapes, and the hexagons are "resultant" shapes, resulting from the adjacency of 3 pyramids.

Geometry is hard!

[Paul Williams]

For the

Pentakis dodecahedron how would you accurately calculate the angle on each piece? Looks like a fun project for a cold winter week.

[Robert Marshall]

Paul asked about the bevel angle(s) for building the Pentakis Dodecahedron.

Good question! Don't know. I expect there's some formula to calculate something like that, but I don't know what it is. I found the bevel angle for the icosahedron from online articles about building them of wood, but I haven't found anything like that for this shape. The calculator page for this shape (https://rechneronline.de/pi/pentakis-dodecahedron.php) doesn't deal with building one out of wood or plywood.

I did find a YouTube video which dealt with building a plywood dodecahedron (https://rechneronline.de/pi/pentakis-dodecahedron.php). It said the bevel angle where you are joining 12 flat pentagonal faces is 31.7 degrees. With a pentakis dodecahedron, the angle of the pyramid would change the bevel (reducing it by the angle of the pyramid vs. the horizontal.)

Good luck. Would love to hear what you find out.

[Barry McFadden]

I think I have to go have a beer..... you guys are giving me a headache with all those fancy words!!!!!