Friday puzzle -- another visualization

Re: Friday puzzle -- another visualization

> and an end face that was a smaller hexagon, twisted 15 degrees from the body.

Are you sure it is 15 degrees?

Re: Friday puzzle -- another visualization

4, 6, or 8 sided.

s/b 30 degrees

Thanks for the catch, Bill. I'd try to claim it was a typo, but even my fingers can't miss that badly! Just a simple brain fart.

No :-(

Re: Friday puzzle -- another visualization

5 or 6 sided (an irregular pentagon or hexagon). Not sure about the second question, but I think the angle needs to be greater than 30 degrees to get a regular shape.

Re: s/b 30 degrees

The best you can do is at least 12 short.

Re: s/b 30 degrees

I'm not sure what you are measuring to come up "12 short", but if that is some measure of inability to make the facets regular, I agree; it can't be done. care to reveal the shape?

Re: s/b 30 degrees

The shape would be a pentagon. A regular pentagon has interior angles of 72 degrees. Your hexagonal prism has interior angles of 60 degrees.

No matter how you cut it, the maximum interior angle at the apex of the resulting pentagonal face will be less than 60 degrees - or 12 short of the required 72.

Correct

The shape would be a pentagon.

Yes. See pic below showing both a sketchup model at 52.5degrees (which gets the right angle where the pentagons intersect with the edges of the original shape, and a picture of one of my actual knobs, cut at 30 degrees.

A regular pentagon has interior angles of 72 degrees.

I'm not sure what you are talking about here. Maybe the angle from one apex to the center to the next apex? What I would call the interior angle would be 108 degrees, the angle at each apex. But maybe my terminology is wrong.

Your hexagonal prism has interior angles of 60 degrees.

Or 120 at each point.

No matter how you cut it, the maximum interior angle at the apex of the resulting pentagonal face will be less than 60 degrees - or 12 short of the required 72.

Now, here you lose me. Assuming I am right that by "interior angle" you mean "Angle subtended by a side from the center", how do you even define a center from which to measure if it is not a regular polygon? I agree with your conclusions, but want to understand your reasoning. I reached that conclusion empirically, by slanting my cut in sketchup until the "bottom" corner was 108 degrees, and then cutting the "top" so that the nearly vertical sides were the same length as the bottom two, and the result was that the top is shorter than the others.

Re: Correct

Nope. Your terminology and angles are correct. I messed up.

It all made sense to my rusty visualizer. But drawing it out on paper, I see that my conclusions were based on totally faulty assumptions.

I'm not much good

at these visulazations--actually, I'm no good at them, but what is this stuff that you are doing called? Solid Geometry? What do you need, computer/software wise to do more of same? Is the learning curve high?

Re: Friday puzzle -- another visualization

I don't think it is possible to get hexagonal facets. Pentagons if there is a flat end of the knob, or quadrangles if they come to a point.

Re: I'm not much good

I guess you'd call it solid geometry. And I am terrible at figuring out solid angles--always have to go back to trig and work with the legs of right triangles to figure out an angle in a plane that is not parallel to one of the primary planes.

The drawing I did was in Google Sketchup. Very good tool for visualizing, free, and very short learning curve compared to CAD software. But that last phase is not saying a lot. I have an old 2-D CAD program that is very easy to use, but I have found most to be unbearably complex for the occasional user. Sketchup is easy to pick up, and even after a period of disuse, I find getting back up to speed relatively quick. I still found this model to be kinda tricky to do in Sketchup, though.

Re: Friday puzzle -- another visualization

I also made an error in thinking about this. I started (mentally) with a square cylinder (square in cross section). Then, taking off the end would result in facxets with a trapezoid or triangle shape, depending on whether the end was a square or point. Then I took off the sides of the cylinder; but of course, this gave me a cylinder with an octagon cross section, not hexagon.

I guess I can say this is an extension of your problem, not a solution.

Thanks. I did the install. Study next.