Is there a name for a three-sided polygon where each side is an arc whose center is the opposite vertex? In other words, imagine an equilateral triangle with curved sides.

A polygon has sides that are line segments. Polygons do not have curved sides. What you describe is not a polygon. (I am assuming we are talking Euclidean geometry here. Otherwise your polygon would be called a “triangle”.)

I just checked the dic, and by definition a polygon must have straight sides. I have never heard a name for the figure you describe, but I am no expert.

If you drew a triangle on a sphere, such that the triangle was large in porportion, it would appear like what you describe…but it would still be called a triangle.

Is this Polygon in Euclidean geometry or hyperbolic geometry?

What the OP is asking for is the name of the shape for the legendary three-sided wheel. This shape rolls smoothly around a central axis, despite its non-circular cross-section. In general, there is no name for such a shape that I have heard of, but some graphics people call figures with such sides “polyarcs,” so I suppose the OP is asking about a particular “triarc.” It’s not only “equilateral” (what would the right word for this be?), but the radius of arc is equal to the chord of the arc. Perhaps it should be named after the first person to describe the shape – does anyone know?

A Reuleaux triangle, or, a constant width curvilinear triangle.

It’s the answer to: What shape, other than a circle, won’t fall through a manhole if you make a manhole cover out of it? Click here.

## Peace.

No, I’m not a genius, but a web search engine makes me look like one.

Equilateral literally means ‘equal sided.’ So, I guess it would be either: equiarckical; equicurvical; or, equiradial.

Peace.

bold mine

I’m almost positive that there are other curves of constant diameter, giving the lie to your statement of uniqueness.

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OK, I checked out the links in this thread, and I gotta say, the bit that can drill square holes is COOL! But I googled “Watts Brothers Tool Works” to try to find where I could get one, and came up empty-handed. Likewise, looking on eBay was fruitless. Any ideas?

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Two British coins are constant-diameter curvilinear heptagons - the 20p and 50p coins, which you can see here.

They were designed this way so that they would be distinctive but would still be recognised by vending machines etc, regardless of the orientation in which they are put in the slot.

Thank you.

I’ve never seen one of these constant-width curve bits+special chuck. The usual tool to make square holes is a hollow chisel mortiser. There are hollow chisel attachments for drill presses (which IMHO are a pain in the butt) and dedicated machines (which are sweet). The bit is a standard auger that rotates within a four-sided chisel. The bit takes out most of the material, and the chisels clean up the sides.

As long as we’re playing Jepardy, It’s also the answer to the question, “what shape is the rotor in a Wankel rotary engine, Alex?”

Yeah, I know about those - I just want to do it the FUN way! A drawing of the special bit was in one of the links further up in this thread.

I have seen an article by Martin Gardner in which these sorts of shapes are referred to as ‘curves of constant width’, and in theory there are an infinite number of them. You can take any regular polygon with *n* sides and curve each side in a uniform way such that you end up with a curve of constant width.

That’s the mathematical answer. Back here in real life, of course, if you asked people to describe the shape referred to in the OP they’d just say “Sort of a triangle, only curvy”. Or, “Like a triangle made of three bananas”.

Or the cross section of a Pepto Bismol bottle.

I wish I’d seen this sooner. I first learned of the Reuleaux triangle in The Book Of Strange Facts & Useless Information by Scot Morris. Since I got the book back in elementary school, I’ve found that some of the information are simply old UL’s. But my favorite piece of information is true- The Spanish word esposa means both wife, and handcuffs.

I may be completely wrong here, but isn’t it called an “involute”?