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Before an undertaking as great as building a Dyson Sphere, it’s a good idea to plan ahead first. This rotating image shows what my plan for an enneagonal-antiprism-based Dyson Sphere looked like, at the hemisphere stage. At this point, the best I could hope for is was three-fold dihedral symmetry.

Augmented 9- Antiprism

I didn’t get what I was hoping for, but only ended up with plain old three-fold polar symmetry, once my Dyson Sphere plan got at far as it could go without the unit enneagonal antiprisms running into each other. Polyhedra-obsessives tend to also be symmetry-obsessives, and this just isn’t good enough for me.

Augmented 9- Antiprism complete

If we filled in the gaps by creating the convex hull of the above complex of enneagonal antiprisms, in order to capture all the sun’s energy (and make our Dyson Sphere harder to see from outside it), here’s what this would look like, in false color (the real thing would be black) — and the convex hull of this Dyson Sphere design, in my opinion, especially when colored by number of sides per face, really reveals how bad an idea it would be to build our Dyson sphere in this way.

Dyson Sphere Convex hull

We could find ourselves laughed out of the Galactic Alliance if we built such a low-order-of-symmetry Dyson Sphere — so, please, don’t do it. On the other hand, please also stay away from geodesic spheres or their duals, the polyhedra which resemble fullerenes, for we certainly don’t want our Dyson Sphere looking like all the rest of them. We need to find something better, before construction begins. Perhaps a snub dodecahedron? But, if we use a chiral polyhedron, how do we decide which enantiomer to use?

[All three images of my not-good-enough Dyson Sphere plan were created using Stella 4d, which you can get for yourself at this website.]

Several days recently swirled down the drain in a depression-spiral. Needing a way out, I spent my Saturday morning playing with triangles, after first getting plenty of sleep. It worked. This technique, however, probably would not transfer to those who are not geometry obsessives. Perhaps any favorite activity would work? I leave that to others to explore.

Here’s what I did that worked for me:

The original triangle is ABC, and is in bold black. The bold blue line is its Euler Line, and contains the orthocenter (M), circumcenter (G), nine-point center (K), and centroid (point W). It does not, however, contain the incenter (S).

It struck me as odd that the incenter would be different in this way, so I investigated it further. It is the point of concurrence of the three angle bisectors of a triangle. On a lark, I constructed the midsegments of triangle ABC, forming a new, smaller triangle, shown in red. When I then found the incenter of this smaller triangle (Z), it appeared to be collinear with S and W. I checked; it was, and this line is shown in bold yellow. Moreover, the process could be continued with even smaller midsegment-triangle incenters, and they were also on this yellow line.

I wondered if I had discovered something new, and started to check. It didn’t take long to find out that Nagel had beaten me to it. The Nagel line is the official name of this yellow line I stumbled upon, and here is my source: http://mathworld.wolfram.com/NagelLine.html — but, as far as I know, I did discover that these midsegment-derived points also lie on the Nagel line.

Someone else may have known this before, of coruse. I don’t know, and it doesn’t matter to me, for I had my fun morning playing with triangles, and now feel better than I have in days.

[Side note: this is my 100th post, and I’d like to thank all my readers and followers, and also thank, especially, those who encouraged me to try WordPress to get a fresh start after Tumblr-burnout. It worked!]