Take a gander at the hypocycloid. You may recognize the shape from the logo of an American football team...

here is their logo

My question is, how do you describe the tip of one such shape in common parlance (in other words, the concave, pointy curve making one-fourth of the example asteroid)?

I know the technical term is "cusp," but in usual literature I take it that cusp means "brim" or "edge," rather than a general concept of "where two curves meet." Further, saying "it looks like a cusp" leaves my imagination staring blankly at nothing (and note that this description must be readable--i.e., I can't just hold my hands in a shape and say "it's like that").

Bonus points for the person who can tell me a word to describe the same shape in 3D (with terms better than an "emaciated cone").

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    You could describe such an edge (inclusive of multiple such "pointy bits") as an inverted scallop edge – FumbleFingers Oct 1 '15 at 16:31
  • My eyes hurt, you need to find a better anti-aliased version of that image. :) – Barmar Oct 1 '15 at 20:31
  • Perhaps "curved triangle", or to avoid confusion with a Reuleaux triangle, you may say a "negatively curved triangle". Another possibility is to replace curved by hyperbolic, e.g. "hyperbolic cone". – Graffito Oct 1 '15 at 22:33
  • I was thinking to describe it like a top (the spinning kind), but knowing the shape of dreidels etc, I knew that wouldn't be sufficient. – Jon Oct 2 '15 at 2:32
  • I'd call it the US Steel emblem. – Hot Licks Nov 9 '15 at 23:40

I would describe that shape as the negative space between circles. If you want the whole shape, between four circles. Two stacked upon two. If you want only the top half of the shape, it would be the negative space between two circles touching side by side, from the line running along the bottom of each to the parallel line crossing the center of the two circles. If you want the 3D shape you would be referring to the cavity in between eight spheres. Four spheres upon four in a matching patterned layer. When you say 'tip' am I to understand you are referring to the yellow shape? Remember when you interpret a visual design that is a graphic image there is always negative space that is the remaining element of the composition. Often in logos the negative space plays an important role when designing for impact. For instance, the FedEx logo has an arrow in the negative space between the 'E' and the 'x' that most people do not notice.

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  • I think by "tip" they mean just one of the pointy bits. Using your terms one could say "the shape of the negative space on one side of the point where two circles meet". – Rupe Nov 10 '15 at 10:26
  • This description has a sesquipedalianness to it. It does describe the form, but only in its perfect, geometrically regular habitat. If I were to stretch the curves (such that we're now talking about two ill-defined ellipses), it would no longer be something that amateur Geometers could envisage. – Jon Nov 10 '15 at 22:06
  • Sorry, I took the question to mean that you wanted a description for the cusps in the picture, rather than those in the general hypocycloid case. – Rupe Nov 11 '15 at 12:31

I believe the term used to describe this in mathematics is "vertex". Google defines it as "the highest point; the top or apex. synonyms: apex, peak, pinnacle, zenith, crown, crest, tip, top".

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  • I believe the OP is looking for something more specific - the shape made by the concave arcs meeting at a point. – Rupe Nov 10 '15 at 10:23
  • @Rupe: Yes, I do think this magical, possibly nonexistent word(s) I'm looking for should be more specific. Since I don't have an image-shack account or anything like that, I thought the Steeler's emblem would give people some indication of what the whole shape looked like, but I'm really interested in describing the emaciated triangle at each of its cardinal directions, somewhat like a carnival tent. – Jon Nov 10 '15 at 22:01

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