# What is the name of a squircle with three, five, six or more corners?

I know that the second shape is a squircle, but what are the other shapes called? Is there an actual name for them?

The shapes do not simply have rounded corners, but the sides have a "circularity". Here is a triangle with rounded corners next to the shape that I do not know the name of:

"Trircle", "Triarcle", and "Pentircle" do not seem to be used for them, at least Google's image search does not provide any results.

• huh, I thought that was called an ovoid. – Lauren-Clear-Monica-Ipsum Oct 18 '17 at 10:13
• I don't know about anyone else, but I'm pretty enamoured of 'Trircle'. I'm adding it to my dictionary. – Easy Tiger Oct 18 '17 at 11:06
• I'd just call it a triangular squircle. – Octopus Oct 18 '17 at 22:49
• Maybe if someone can generalize and give these functions (on Math.SE) a name, then we have the answer... – Andrew T. Oct 19 '17 at 4:57
• The middle one is 50p last one is 20p. – Strawberry Oct 19 '17 at 15:19

"Squircle" was a random mash-up someone somewhere came up with and it became trendy. But a square with rounded corners, is still a square. And a circle with any corner is no longer a circle.

There are no specific names for the shapes merely because they have rounded corners. A triangle is still a triangle regardless of how round the corners may be. The defining factor is the number of sides, not the corners.

Now you can try and start your own trend the way "squircle" is a trend.... invent your own names.... then use them constantly, repeatedly, in every way you possibly can. Maybe they'll catch on.

• A Squircle is not simply a square with rounded corners. – Wrzlprmft Oct 18 '17 at 9:20
• "A triangle is still a triangle regardless of how round the corners may be." To a geometer, a triangle is a shape whose boundary is exactly three straight line segments and nothing more: if it has rounded corners, it's not a triangle. – David Richerby Oct 18 '17 at 12:09
• More to the point, the word is tri-angle which kind of does emphasize the corners. Likewise with rect-angle, quadr-angle, and also more generally with poly-gon, penta-gon, hexa-gon, and so onâ€” gon comes from the Greek for angle or corner. – KRyan Oct 18 '17 at 15:00
• Yes it's possible to be pedantic. However, this is art, not mathematics. – Scott Oct 18 '17 at 15:20
• I donâ€™t think itâ€™s pedantry at all. As indicated in the question, a triangle with rounded cornersâ€”on the left in the second imageâ€”is quite distinct from a â€śtrircle,â€ť on the right. If you say to me that you want me to put a â€śtriangle with rounded cornersâ€ť in a design, you are going to get the shape on the left, and definitely not the shape on the right. These shapes do not merely have rounded corners, thereâ€™s more to them than that. – KRyan Oct 18 '17 at 15:46

Well, its true that a rounded triangle works. Except the sides are also not straight so you wouldn't know also sides are rounded. However there is a mathematical shape that exhibits this kind of form. And that is a Epitrochoid.

Image 1: a suitable set of Epitrochoid.*

Therefore we could thus call these shapes

• 3 lobed Epitrochoid
• 4 lobed Epitrochoid
• etc.

However, Epithorcoids include quite a lot of other shapes too, so for example even the adobe logo is a 3 lobed Epitrochoid. Realistically speaking though we can not have a name for all shapes. So let us describe them instead of name them all.

Image 2: a unsuitable set of Epitrochoids

* code used in Mathematica: Table[ParametricPlot[{Sin[t - o] + 0.3/(lx) Cos[lt - o], Cos[t - o] + 0.2/(lx) Sin[lt - o]} /. {x -> (l - 2)*0.2 + 1, o -> [Pi]/(2 + (l - 2)*2)}, {t, 0, 2 [Pi]}, Axes -> False], {l, 2, 7, 1}]

• +1 either way, but I think this answer would be improved by the actual definition of epitrochoid, as well as any possible descriptors we might attach to indicate the ones we want (e.g. does the term convex epitrochoid exist, and if so does it mean what we want?). – KRyan Oct 18 '17 at 21:04
• @joojaa not Hypotrochoid? – martin Oct 19 '17 at 11:51
• @martin Well in this case its a epi but i guess yo get same shapes can be generated with Hypotrochoid OR in fact any rotation on rotation family of shapes. – joojaa Oct 19 '17 at 15:36
• Since Epitrochoids are based specific equations, I think this term is too narrow. The shapes in the original question are no Epitrochoids, because they were created using a different algorithm (I happen to know because I wrote it). While your answer is great, I think another, broader name should be used that also represents similar shapes created with a different algorithm. – Waruyama Oct 20 '17 at 8:33
• @Waruyama i didnt actually suggest this as a name i just said that a shape that fulfill the critertia and they have names. – joojaa Oct 20 '17 at 12:03

Answering after doing a bit of research prompted by a comment by Waruyama.

Referring to these as Reuleaux polygons, e.g. Reuleaux triangle, might get you somewhere. These polygons are much closer in appearance, to my eyes, than polygons with rounded corners (which are, to me, quite distinct and not a sufficient description of these shapes at all). However, the term has a number of issues:

• It is not well known outside geometry and specific technical fields (theyâ€™re used in some engines, for example), and the name doesnâ€™t hint at anything.

• Reuleaux polygons are very specific mathematical shapes with particular properties. You cannot simply take a polygon, curve the sides a bit, and claim it is a Reuleaux polygonâ€”that would refer only to a polygon with very specific curves to the sides.

• Only polygons with an odd number of corners can properly be called Reuleaux polygons. So a squircle cannot be a Reuleaux polygon, no matter how carefully you curve the sides.

• And for that matter, those corners are sharp, not rounded. Though saying â€śReuleaux polygon with rounded cornersâ€ť might get you around that one.

• Finally, it appears that there is a company called Reuleaux that sells paraphernalia for vaping, and that tends to dominate search results, which will cause problems in understanding and discoverability.

Reading the linked Wikipedia page does offer a link to circular triangle, however, and that term has much greater promise: it is a general term for triangles formed from circular curves. The Reuleaux triangle is one, but this term can also cover a variety of other shapes. In fact, it can cover shapes that we would not consider the same as your â€śtrircle,â€ť since the curves that form it can be convex or concave. In these figures they are all convexâ€”which can be communicated, per the article, with â€śconvex circular triangle.â€ť

Since we are also not being very picky about our curvesâ€”they are not necessarily circular curves, reallyâ€”we can generalize that term too. AAGDâ€™s answer suggests â€śconvex elliptical triangleâ€ť where an ellipse is a more general term for curves that include circles, so that is a step in the right direction, but then we also really arenâ€™t necessarily referring to elliptical curves either (and this can also run into some confusion with elliptic geometry, which again look similar but are not quite these shapes).

So I am going to suggest that we could use the term â€śconvex-curve triangles,â€ť and more generally, â€śconvex-curve polygons.â€ť Probably â€świth rounded corners.â€ť That would cover precisely the shapes in question.

Itâ€™s also basically unheard of. Google finds 6 results for `"convex curve triangle"`. One is selling jewelry with stones cut into the appropriate shape, and another appears to be an art gallery with a geometric bent, and both are using the term to refer to the â€śtrircle,â€ť so at least we arenâ€™t contradicting what little prior usage there is, but thatâ€™s not saying much. `"convex curve polygon"` gets 10 results, but they all appear to be highly-technical geometry research papers.

Finally, I would note that the term that was most accurate for these shapes while still being within the realm of â€śpeople actually use this termâ€ť was â€ścircular polygons,â€ť from which we can clearly see the actual derivation of squircle: square-circle became squircle. Likewise, triangle-circle becoming trircle, pentagon-circle becoming pentircle or pentarcle or something, and so on. So while these names are not frequently used, as noted in the question, they are both accurate (as shortenings of the â€ścircular polygonâ€ť terms) and a clear extension of the better-known â€śsquircle.â€ť So my conclusion, ultimately, is to echo filipâ€™s answer, and suggest that these names are the best choice for regular use.

Trircle, Squircle, Pentircle, Hexircle, Septircle? No, they probably have no names. Personally I would call them "triangle/square/... with rounded corners".

• I have edited my question to show the difference between shapes with rounded corners and the shapes that I do not know the names of. – Waruyama Oct 18 '17 at 7:51
• A squircle is not the same as squares with rounded corners (or a rounded square). It is a shape of its own, a kind of superellipse. – H.W. Sanden Oct 18 '17 at 7:52
• After digging a bit the "Trircle" could be described as a combination of a rounded triangle and a Reuleaux triangle en.wikipedia.org/wiki/Reuleaux_triangle – Waruyama Oct 18 '17 at 8:01
• I found a patent application that refers to the trianular shape as a "Reuleaux triangle with rounded vertices" google.com/patents/EP1127019B1?cl=en – Waruyama Oct 18 '17 at 8:22
• Likely all these shapes are Gielis curves that can be generated by Johan Gielis' en.wikipedia.org/wiki/Superformula. You can try the formula here: procato.com/superformula – AAGD Oct 18 '17 at 18:33

elliptical convex triangle, elliptical convex pentagon, elliptical convex hexagon and so forth...

As much as I like the word "squircle", I think that fitting the other shapes into an "ircle" would quickly get out of hand; additionally it feels like a very esoteric term.

May I suggest puffy-rounded-triangle/square/pentagon/hexagon/heptagon/etc.â€Śâ€‹..? This way, the average Jane/Joe can understand what you are talking about as well.

The term 'squircle' is understood because there is enough of each of the component words remaining, and it's caught on as it's nice and short, and fun to say. The same can't be said for trircle and the other contractions after that style.

A common way to distinguish between members of a family that differ only in some number of something, at least in mathematics, is to use a numeric prefix.

My name for the three sided version would be a 3-squircle.

Part of the benefit of this technique is that I know everybody reading this answer, without exception, will be able to construct the unique name of any other rounded polygon squircle-like shape, regardless of the number of sides.

There is obviously a glaring inconsistency. A squircle has 4 sides. However, the fact of that inconsistency indicates that we are using the term squircle in a different but related way, to describe the family of shapes, rather than the precise shape. The '3-' prefix, being so clear, obviously overrides the implied order of the shape.

The numerical inconsistency, and the fact that it's glaring, also inject a bit of levity into the name, it's fun.

If you were communicating about your design, you might use the term 4-squircle at some point, to emphasise its slight shift in meaning.

Once the term squircle has been freed of the need to communicate the order of the shape, perhaps a new shape name could be constructed, like polyround, or circlegon - remember it's got to be a single word, not too many syllables, syllabic stress compatible with being easy to say, with roundness and sidedness clearly implied - a tough ask. So would I use '4-polyround' over 'squircle' or even '4-squircle'? I think not. 'Sidedround'? Maybe ... not. 'Roundygon'? Hmmm, maybe.

I am afraid it gets worse than you think

# The shape you indicate is technically not a squicle

According to wikipedia, a squircle must exactly match this formula:

(x-a)^4+(y-b)^4= r^4

Unless my eyes deceive me, the sample image that you provided does not exactly match this equation.

Hence unfortunately:

### We must fall back to a more generic description

This is merely an attempt, it could still be refined:

"Shape" with rounded edges and curved sides

For instance

Equilateral triangle with rounded corners and curved sides

Or perhaps a bit narrower (not sure if it fits, but optically this seems to be accurate):

Convex circular equilateral triangle with rounded corners

we have run into this problem when we discuss Voronoi patterns and issues surrounding manufacture and biocompatibility - we have used the terms "circangle" for circular triangles, and "circazoids" for "circular trapezoids" - right or wrong