What are the Spherical Interior Angles for Regular and Archimedean Solids?

[RandallB]

Anybody know where to find a listing of the Spherical Interior Angles for Regular and Archimedean Solids.

For example on a Buckyball (Archimedean Solid - truncated dodecahedron) should have angles:

A: Angle between vertexes (length of Edges in radians)

B: Angle from center Hexagon to edge

C: Angle from center Hexagon to vertex

D: Angle from center Pentagon to edge

E: Angle from center Pentagon to vertex

That gives: A + 4B +2D +2E = Pi (or 180⁰)

Just haven’t been able to find a referance that has this kind of detail.

 

[robphy]

Possibly useful: http://www.intent.com/sg/polyhedra.html

 

[Doodle Bob]

That gives: A + 4B +2D +2E = Pi (or 180⁰)

Just haven’t been able to find a referance that has this kind of detail.

This might be Descartes' Angle Deficiency Theorem in disguise. The angle deficiency of a polyhedron at a vertex is: 2*pi - sum of the angles of the faces intercepting that vertex. Descartes Theorem (which is provable relatively easily from Euler's Formula and which is the discrete version of The Gauss-Bonnet Theorem) says that the sum of the angle deficiencies of all of the vertices of a closed polyhedron is always 4*pi. You're adding up a bunch of different angles above, so that might account for the difference.

Anyway, some good references. First of all, look up any high school text from before 1950 with the terms "Solid Geometry" and/or "Spherical Trigonometry" in their titles. I like the one wirtten by Osgood and Graustein myself. You can find scads of them on Ebay or probably your local used bookstores. Buy as many as you can, since no one seems to publish this kind of stuff any more.

Another handy reference is Polyhedra by Peter Cromwell (currently published) and the CRC Handbook of Convex Geometry, which has a big section on polyhedra and rigidity.

If you're not up for a paper-chase, I would suggest you start with Mathworld first: www.mathworld.com, and look up Solid Geometry.

 

[RandallB]

That’s better than I’d already seen at mathworld. But most references seem to stop at dealing with the edge, inradius, midradius, and circumradius details. I was hoping to find a reference to give the interior angles directly without too much calculating on my own required.

Didn’t think it would be this hard to find a source with a table that had already ground through the math. I keep looking, maybe an old book is the place to find it.

-- Don’t know “Descartes' Angle Deficiency Theorem” is
I was just adding up the individual angles to go half around a soccer ball.

 

[Doodle Bob]

I was just adding up the individual angles to go half around a soccer ball.

Upon rereading your original posting, it's slightly unclear what angles you are calculating. Most texts deal mostly with solid angles (which are devilishly hard to define) about the vertieces and dihedral angles about the edges. In fact, most other angles can be calculated from knowledge about these.

Nevertheless, this type of calculation was often the type that homework problems would center around in those old textbooks. So, you might have some luck there. Don't bother looking for "Buckyball" references, though. That term, i believe, was invented in the 80s.

 

[RandallB]

Upon rereading your original posting, it's slightly unclear what angles you are calculating. Most texts deal mostly with solid angles (which are devilishly hard to define) about the vertieces and dihedral angles about the edges.

I’m talking about the interior angles as measured from the center of the sphere, not the angles about the vertex’s or any other points on the flat planes of the polygons that make up the surface of the solid polyhedron.

Think of it as looking for the distance on those flat polygons between the various vertex, centers of the edges, and centers of the polygons, but measured in radians rather than normal length.
That would represent the spherical shadow cast on any size sphere for those lengths. In this way you have the separation between the points that remains the same for all three spheres related to the polyhedron instead of dealing with three different numbers for the in-sphere, mid-sphere, and circum-sphere.

What we’ve found so far are tables of traditional lengths based on in-radius, mid-radius, and circum-radius. Rather than giving them in the simple uniform measure of radians.

 

[Doodle Bob]

Hmmm... I see what you're doing now. I must admit that most sources that I've seen stay within the traditional dihedral angle measurements. Some thoughts, many of which you probably know already:

1. Keep in mind that in general a convex polyhedron will not have any unique spheres attached to them, e.g. a unique circum-sphere, etc. I don't know the context in which you're studying these things, though, so that might not be a concern.

2. If you're restricting yourself just to Platonics and Archimedeans, since most of the Archimedeans are truncated versions of Platonics, just computing the interior angles of the Platonics might save a lot of work.

Otherwise, I'm afraid I can't help you out any more than that. Here are some more books that you may find useful.

Regular Polytopes by Coxeter
Introduction to Geometry by Coxeter
Polyhedra Primer by Pearce and Pearce
Convex Figures and Polyhedra by Lyusternik

 

[RandallB]

1. Keep in mind that in general a convex polyhedron will not have any unique spheres attached to them, e.g. a unique circum-sphere, etc. I don't know the context in which you're studying these things, though, so that might not be a concern.

I understand, in fact I guess that’s the point. If the surface shapes are defined in lengths of radians there is no such thing as a different or unique circum-sphere to compare between polyhedrons. By using radians it puts them all on a common measure for comparing surface lengths and areas; even interior volumes (All based on radians projected onto the surface of any sphere).

It’s as if all the descriptions of this three dimensional geometry are only cataloged in two dimensional terms, rather than applying a 3-Dimensional spherical description.
Seems to me comparing these 3-D things would be much more clear if a better language was used.
I’m just a little surprised that generating these kind of measurements for such fundamental shapes hasn’t already been done by someone.

 

[Doodle Bob]

It’s as if all the descriptions of this three dimensional geometry are only cataloged in two dimensional terms, rather than applying a 3-Dimensional spherical description.
Seems to me comparing these 3-D things would be much more clear if a better language was used.
I’m just a little surprised that generating these kind of measurements for such fundamental shapes hasn’t already been done by someone.

I'm quite sure that these measurements have been done before. There are quite a few reasons why people measure the three-dimensional objects the way they do. 3-diml. measurement, such as via solid angle, is notoriously persnickety.

For example, two angles in the plane with the same measurement are congruent to each other. There is some rigid motion that moves one angle right on top of the other. But two cone-like objects with the same solid angle (i.e. using some sort of "fraction of the surface area of a sphere" measurement) are almost never congruent.

However, two pairs of planes intersecting at the same dihedral angle are indeed congruent. And in fact I believe that two solid angles coming from polyhedral cones with congruent corresponding dihedral angles are indeed congruent.

 

[RandallB]

I'm quite sure that these measurements have been done before.

Web being what it is, you’d think someone would have those listed somewhere.

But two cone-like objects with the same solid angle (i.e. using some sort of "fraction of the surface area of a sphere" measurement) are almost never congruent.

I don’t understand what your saying. Are you calling “solid angle” an Interior Angle measured from at the center of a sphere. Any such angle would define a single line (edge) on the surface of any sphere (three or more lines needed to define an area). Which would have to be congruent with any measurement of the same angle. They could never not be congruent.
Even if the spheres are of different sizes the shadow of the inner line onto the larger sphere would exactly match the 'larger' line if they have congruent Spherical Interior Angles.

I think your describing the problem of comparing lines on planes that cut though depths of a sphere at different ways. That problem doesn’t exist if you measure the lengths in radians. That’s the point of making the comparisons in radians.

 

[Doodle Bob]

I think we're talking about two different objects. By a solid angle, I mean the convex hull of a set of rays with the same endpoint, e.g. a cone, a pyramidal cone. the most common way to measure this "solid angle" is to construct a ball of radius 1 with center at the endpoint of the rays. The measure of the solid angle, then, is the area of the intersection of the solid angle with the boundary of the ball, ie. the 2-sphere.

Clearly, two of these objects can have the same measure as above and not be congruent. (Note that congruence means that one of the objects is the image of the other under a rigid motion).

Anyway, this is obviously different than what you have in mind.

 

[RandallB]

Anyway, this is obviously different than what you have in mind.

No kidding – way off subject.

For those just coming to this thread, what is being looked for here is some reference that provides detail on the size of lines on the surface of the 17 Regular and Archimedean Solids; BUT measured by the subtended angle from the center point of the solid for those lines!

Measured in radians this will also be the length of the shadow of that line projected onto a unit radius sphere using the same center point.

Most locations like, mathworld or wikipedia, at best only give this for the edge lengths. Not for other lines drawn from the center of the various flat surfaces to their vertexes and edge centers.

I see a lot of value in comparing the solids by using a uniform measure like this.
So if anyone knows of some place that has already collected this kind of detail please advise.