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I really like the logo at Worldbuilding SE; it features a polyhedral planet and a beautiful city with a flying whale ship. That planet is particularly neat. Light glances off its facets, resulting in what would be called "fire" in a giant gem; scattered light scintillates orange and green and overlays of both. It's beautiful and interesting.

But could it exist?

It appears to be a Icosidodecahedron (https://en.wikipedia.org/wiki/Icosidodecahedron), with one round of triangular infilling applied. If you kept subdividing those triangles into more triangles, you eventually end up with a shape indistinguishable from a sphere.

Now, we all know spherical planets can exist; in fact, posting questions positing worlds that are squares, donuts, or, in the case of one special case of idiot, a banana, receive fairly strident criticism for ignoring the effect of gravity. All the same, Earth has "small" irregular features like mountains that are not eliminated by gravity, not in any directly observable timeframe, anyway.

My question is: Let's assume Logoworld (the Weltlogo?) has earthlike mass and composition but no water. Can the planet exist in its depicted shape - an icosidodechedron with one degree of triangular infilling? Or will gravity eliminate the facets? The edges need to be sharp enough to be visible from the moon under optimal conditions, with the naked eye.

Weltlogo

Please ignore: 1) Rotation and/or other secondary effects that distort the planet by a percent or two, 2) Logoworld's ring. 3) Atmosphere, oceans, erosion and other planetary features other than gravitational rounding that urge you to shout FRAME CHALLENGE!, 4) The floating whale ship and its city, 5) the orbiting capital letters. EDIT: No explanation of the origin of the facets is necessary, let's take it that it was made that way deliberately by its creator.

These pages may be helpful: Truncated Icosahedron (https://en.wikipedia.org/wiki/Truncated_icosahedron#Orthogonal_projections)

Geodesic polyhedron (https://en.wikipedia.org/wiki/Geodesic_polyhedron)

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    $\begingroup$ By looking at the tiny shadowy nuances here and there, it looks more like it's more all triangles than a mix with pentagons. It's further hinted if the golem is projecting the same planet as the logo... $\endgroup$ Jun 28, 2022 at 10:47
  • $\begingroup$ @Tortliena, agreed there's no actual pentagons; one degree of triangular infilling has subdivided any polygons that were there (if they were). I had never noticed the golem's green haired girlfriend before. $\endgroup$
    – user86462
    Jun 28, 2022 at 10:55
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    $\begingroup$ Say what you will about the civilization that built a world just to be part of a giant sign that spells “WORLDBUILDING” - they have an impeccable sense of metahumor. $\endgroup$ Jun 28, 2022 at 11:46

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Logoworld is a body that is pretty spherical and quite closely to a perfect sphere. It is not an Icosahedron, which has only 20 sides, it is a sphere faceted to show us about 50 facets. So it has at least 100 faces, very much akin to "D100" though that has (mostly) pentagonal facets, not triangular ones, as this commercial product shows: a commercial d100

However, a d120 would have triangular faces, yet those would be irregular and not spherical. The point of the argument is not to denominate a specific number of faces but show that it could be done to triangulate a sphere. Or rather, create an "Ico-sphere". The next picture is a 3-subdivision ico-sphere from blender. This object has 320 faces, 480 edges and 162 vertices.

At about that level, the divination from a true sphere starts to get small enough percentagewise that hight differences become more the order of magnitude of mountain ranges. The more faces we add, the more our ico-sphere becomes an approximation of a true sphere: a 6-subdivision ico-spehre with 10242 vertices and 20480 facetts is basically indistinguishable from a true sphere. To show, the 320 face and 20480 faces ico-spheres next to one another:

enter image description here

We can even convert this into a 2D problem: any cut through the body will give us an inscribed or circumscribed n-gon to a circle. at n=16, the difference between the two is about 2.5% from the real area (pi). At above n=17, the difference between the real, inscribed, and circumscribed areas get even smaller and shrink even further. There are math programs to solve such. At n=20 we almost have a sphere visually and only differ from it by less than 1 percent (0.8 % crosssection difference to a true sphere, ). At about n=50, our solution is very close to Pi - we are at one permille difference now.

Plot of the circumscribed area

It gets easier to see if you actually divide by the circle area and get the percentage itself.

enter image description here

As a simple approximation, we'd get about n² faces by placing our n-gons, so we expect somewhere between 320 and 400 faces to well enough approximate a spherical planet where the height differences fall off to neglectable in a small scale, while large planets might need a bigger number n.

That 20480-face ico-sphere? That's equator is basically a 143-gon, with only about 0.018 % difference from the equatorial area a real sphere should have. If we put in earth as the base, each of the peaks is about one Mt Everest from the sphere surface.

Solution caveats

A sphere with triangulated facetts would get us close enough to spherical to be allowable. However, there are four caveats:

  • The body has to be tectonically inert, so the neat facet boundaries will not get shifted or destroyed.
  • Logo-World was extensively terraformed to achieve the neat boundaries, especially keeping woods contained and facet ridges sharp.
  • The number of facets might need to be increased to more than 100, possibly closer to thousands, but then it would become very stable.
  • The planet lacks oceans, as those would violate the sharp ridge structure.

However, I can give you the ring: the presence of a ring is not a problem, even for a rocky planet.

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  • $\begingroup$ With respect, Trish, and it is, because I've seen you give good answers, I submit that you have not got this one right. Upvoted anyway though. I think Logoworld is indeed an Icosidodecahedron (not icosahedron) with one round of triangular infilling, as described. It's not a D100; you can see clearly that the facets are triangular. I'm going to explicitly list oceans along with the other secondary features, although I think it's implicitly excluded already. $\endgroup$
    – user86462
    Jun 28, 2022 at 8:36
  • $\begingroup$ @SeanOConnor Oceans can't be delivered - they'd not only erode the trigons, but also they would not confirm to the flat surfaces. Which is why I banned them. $\endgroup$
    – Trish
    Jun 28, 2022 at 9:07
  • $\begingroup$ yes, which is why I'm now excluding them explicitly as well as implicitly. Erosion was excluded explicitly from the outset. The real question is gravitational rounding. $\endgroup$
    – user86462
    Jun 28, 2022 at 9:13
  • $\begingroup$ @SeanOConnor 320 faces ico-sphere gets the diviation small enough. $\endgroup$
    – Trish
    Jun 28, 2022 at 9:15
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    $\begingroup$ Going to 2D is a good idea. Using keisan.casio.com/exec/system/1223432608, with 18 sides, it looks like you have to increase the circumradius of an 18-gon by a factor of 1.01 to get a polygon/ unit circle area ratio of 1.00. I think (?) with earth's radius of 6400 km that implies a 64 km protrusion? I'm not convinced 18 sides is enough. $\endgroup$
    – user86462
    Jun 28, 2022 at 10:38

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