ndl wrote:quickfur wrote:Mainly, my interest was driven by trying to understand what exactly "longdome" is.

I also got interested in crinds recently and rediscovered this "longdome". I would call it a "Crindal Cylinder" maybe in following with the names on the wiki. I created vertices for an approximate of the shape and loaded it into Stella4d. I posted a picture here. If you want I can send you the .off or .stel file.

Basically I'm just trying to understand exactly how this "longdome" or crindal cylinder is constructed. After reading through Marek's post carefully, I understand now what its projection envelopes are, but still have trouble visualizing its structure. Hence the question about tricylinders, because depending on how you define the operation used in the derivation, you may end up with different shapes.

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The tricylinder is very interesting because it brings out the following question: What type of operation are you looking for? A crindal operation, so to speak, takes an existing n-dimensional shape and reduces it to a point in the n+1 dimension on both sides. But there are 2 ways of doing that reduction, it can keep the entirety of the shape and scale down the size, or you can truncate it on all sides. As you pointed at those are both interesting shapes.

It's not just 2 ways of doing it; there's also the question of how the scale varies over the (n+1)'th axis. If it varies linearly, then you have a bipyramid. If it varies sinusoidally, then you have a crind-like shape. But it can also vary by an L shape (right-angled), which gives you a cylinder or prism, or by an obtuse-angled L shape, which gives you an elongated bipyramid, or by a hook shape (circular arc + line segment), which gives you an elongated crindal prism, and so on. The possibilities are infinite.

And as you said, you can truncate it on all sides, which at first glance seem to be a parallel to the various ways of bevelling / Stott-expanding a polytope. For example, in 3D, you can start with a square at z=0, then truncate its vertices linearly as z varies, until it becomes a point at z=±1. That gives you a polyhedron with 4 rhombuses connected tip-to-tip in a square configuration, capped with two florets of 4 rhombuses that meet at a point, sort of like a modified dual square antiprism where the two halves are separated, then rotated to be joined vertex to vertex, with 4 new faces inserted to fill in the gaps. If we use a sinosoidal variance instead of linear, we'd end up with 4 circles around the equator, with a 4-legged curved dome shape capping each side, sorta like the intersection of a sphere with an AxAxB cuboid where A < radius and B > radius.

But in higher dimensions, there are many more ways to "truncate on all sides"; for example, starting with a cube embedded in 4D, we could taper it to a point on w=±1 either by scaling, which gives either a bipyramid (linear) or a crind-like shape (sinusoidal), or by truncating its corners, or by truncating its edges, or by truncating both vertices and edges. Each way produces a different shape.

Furthermore, even within the same kind of truncation, e.g., truncation of vertices, there's the question of how the vertex is truncated: by a straight edge, as assumed above, or by a curve? E.g., we could start with a square, implicitly inscribed in a circle, then truncate it by shrinking the circle and taking the common intersection. Depending on how you vary the radius of the circle, various different kinds of shapes could be obtained.

And if you wanted to produce even stranger shapes, consider what might happen if the circle is shrunk not by reducing its radius on all sides, but only along one axis, i.e., it becomes an ellipse, narrower and narrower until it ends in a line segment. In fact, you can start with a circle, and consider its sweep along the z axis as it shrinks along one dimension into ellipses and eventually a line segment. You'd get a kind of cone-like shape with two apices instead of one. Or if you shrink it sinusoidally, a kind of distorted hemisphere with two sharp corners.

The possibilities are endless.