# 17 Geometric Shapes You've Never Heard of

Most people consider themselves smart for knowing what a Möbius strip, icosahedron, and a hendecagon are, but unless they are mathematicians or designers, they've probably never heard of most of the unusual and rare shapes described below. (And if you don't know the strange geometric shape names above, see the end of the article.)

If you want to impress your friends at parties and increase your trivia knowledge by 0.001%, learn this list of obscure geometric shape names. You may actually know some of these figures by more informal terms rather than by their proper geometric names. Or the next time you substitute teach a kindergarten class, forgo the standard lesson on squares and triangles and blow their little minds with these amazing shapes instead. (And don't forget to take the poll!)

## (1) Selburose

A selburose is a type of octagram, an eight-pointed star formed by joining every third vertex of a regular octagram. They are a traditional knitting pattern from Selbu, Norway and commonly seen on sweaters and mittens.

## (2) Stella Octangula

The stella octangula, also known as the stellated octahedron, can be considered as two regular tetrahedra intersected so that their edges cross at perpendicular angles. It can also be constructed by starting with a regular octahedron and affixing eight regular tetrahedra to each of the eight equilateral triangular faces.

## (3) Lune

A lune is a crescent moon shape whose two arcs are circular arcs. If the curved arcs of a crescent moon shape are something other than circular, then it isn't technically a lune. Lunes can be thick or thin, and the two circles used to make the shape can be the same size or different sizes. Various lunes are shown in yellow below.

**Bonus Fun Fact:** If you construct a pair of equal lunes by taking two circles of the same size and overlapping them so that the circumference of one lies on the center of the other, and vice versa, the resulting shape between the lunes has an even more specific name, a *vesica piscis*, which is Latin for bladder of a fish. In general the football shape between two overlapping circles of the same size is called a lens. The next time you see a Venn Diagram, you can think of it as two lunes and a lens.

## (4) Klein Bottle

A klein bottle is a closed curved surface whose "outside" and "inside" surfaces are the same. It's the 3-dimensional version of a Mobius strip.

## (5) Borromean Rings

Borromean rings are a set of three linked rings arranged in such a way that if any one ring were to be removed, then the whole thing would become unlinked. In other words, no two rings are linked to each other, yet as a set of three they cannot be separated.

## (6) Balbis

A balbis is actually not such a strange and unusual shape after all, in fact, you've probably known it since kindergarten. "Balbis" is nothing more than a fancy name for the shape of a capital H (and capital I depending on the font) and also for a rectangle with one side missing. The technical definition of a balbis is a line segment whose endpoints intersect perpendicularly with another pair of lines (whose lengths may be infinite). The origin of the word is Greek and means a rope mounted between two vertical posts to mark the start and finish of a race. Some examples of balbises are shown below.

## (7) Astroid

An astroid, not to be confused with the space rocks called asteroids, is a four-pointed star-like shape with four concave curved "sides." It is defined by the equation y^(2/3) + x^(2/3) = C, where C is some constant greater than 0. A stretched astroid defined by the equation (Ax)^(2/3) + (By)^(2/3) = (A^2 - B^2)^(2/3) is the evolute curve of an ellipse with major axis length 2A and minor axis length 2B.

An astroid can be generated by taking a circle of radius R and rolling it all the way around inside a circle of radius 4R.

## (8) Deltoid

A deltoid is similar to an astroid, but with three points and three inward curving "sides" instead of four. A deltoid can be formed in a similar manner to the astroid as follows. Take a circle of radius R and mark a point on its circumference. Roll the circle of radius R inside a larger circle of radius 3R and trace the path of the point on the smaller circle as it rolls. You can graph a deltoid with the pair of parametric equations

x(t) = 2cos(t) + cos(2t) and

y(t) = 2sin(t) - sin(2t)

## (9) Salinon

A salinon is a curved planar shape whose boundary is a four half-circles arranged in a particular way.The name is from the Greek for salt cellar. If the top part is a semi-circle of diameter D, then the bottom part is formed by three semi-circles of diameters A, B, and C such that A + B + C = D. If the bottom three half-circles are in the order A, B, and C, then the outer two A and C have the same direction as the top one D, while the middle one B goes in the opposite direction. The width of a salinon is D and the height is (D+B)/2. Curiously, if A = C, then the area of the salinon is the same as the area of a circle with a diameter of (D+B)/2. Some examples of salinons are shown below.

## (10) Arbelos

An arbelos is similar to a salinon except that it is made up of three semi-circles instead of four. If the top part is formed by a half-circle of diameter of D, then the bottom is formed by two half-circles of diameters A and C such that A + C = D. All three semi-circles are oriented in the same direction. Given A and C, the area of an arbelos is π*A*C/4 and the perimeter is π*(A+C). Like many shapes, the origin of the name is Greek, meaning a shoemaker's knife.

## (11) Reuleaux Polygons

Reuleaux polygons are not true polygons since their sides are circular arcs rather than straight lines. To be a reuleaux polygon, the shape must have a constant diameter. Diameter is defined as the distance measured from one vertex to the opposite curve, which means that all reuleaux polygons are odd-sided

An equilateral reuleaux triangle has sides that are 60-degree circular arcs. If you stretch a string from one vertex to any point on the opposite arc, the length remains unchanged. A reuleaux pentagon has sides that are 36-degree circular arcs. Because Reuleaux polygons have the constant width property that circles have, they can be used as wheels. See photo and video of a Chinese man who made a bycicle with a Reuleax triange and Reuleaux pentagon as wheels.

## (12) Superellipse, Lamé Curve, Squircle, Squoval

Rectangular and square-like shapes with rounded sides, or circular and oval-like shapes with four flattish sides are called superellipses, and are defined by mathematical equations called Lamé curves. The colloquial terms for these shapes are squircles and squovals, portmanteaus of square and circle, and square and oval. The euqation of a Lamé curve that produces a superellipse is

|x/a|^k + |y/b|^k = 1

where a and b are the half-length and and half-width of the shape, and k is a number greater than 2. If k = 2, the shape is a true circle or true oval. If k = 1, the figure has straight sides and the shape is a diamond or rhombus. If k = 2/3, the resulting curve is an astroid (see #7 above). In general, for an arbitrary value of k there are no closed-form formulas for the perimeter and area of a superellipse, but they can be approximated with numerical methods. Here are some superellipses with different equations graphed together on the same plane.

## (13) Truncated Icosahedron

A truncated icosahedron is a polyhedron with 12 regular pentagon faces and 20 regular hexagon faces, 60 vertices and 90 edges. The pentagonal faces are arranged like a dodecahedron. You may know this shape by its common name "soccer ball" or if you are a chemist, by the name "buckminsterfullerene" or "buckyball," an allotrope of carbon made of 60 carbon molecules arranged as the vertices of a truncated icosahedron. It gets its name by taking an icosahedron (solid geometric shape with 20 equilateral triangular faces) and cutting off the tips of all 12 vertices so that the stumps are regular pentagons and the once triangular faces become hexagons.

## (14) Triskele, Triskelion

A triskele or triskelion is an ancient symbol made of three connected or interlocking spirals that swirl in the same direction. The flag of the Isle of Man is a triskelion stylized as three legs. The triskele is an ancient European symbol and the emblem of Sicily in Italy, the Brittany region of France, the Isle of Man in Great Britain, and Essen in Germany.

## (15) Rhombic Enneacontahedron

The rhombic enneacontahedron is a 90-faced polyhedron whose faces are all rhombuses. It has 60 wide rhombic faces and 30 narrow rhombic faces. The wide rhombus have a diagonal ratio of 1 : sqrt(2). The narrow rhombuses have a diagonal ratio of 1 : φ^2, where φ is the Golden Section equal to (1 + sqrt(5))/2, and φ^2 = (3 + sqrt(5))/2 = φ + 1.

## (16) Ogee

An ogee is a shape made of S-shapes, often used in Islamic architecture to make arches, and in tessellation or tiling patterns that look like stylized lemons or onions. The curly bracket symbol { could be called an ogee. Here are some examples of ogees in arches and wallpaper design.

## (17) Lemniscate

The lemniscate is a closed curve that intersects itself once and resembles an 8 or infinity (∞) symbol. There are actually several unrelated mathematical equations that produce figure-eight shaped curves. The Lemniscate of Bernoulli is given by the implicit function equation

x^4 + y^4 + 2(xy)^2 - Cx^2 + Cy^2 = 0

where C is some constant that governs the size of the lemniscate. The Lemniscate of Gerono or Lemniscate of Huygens is given by the simpler implicit function

x^4 + y^2 - Cx^2 = 0

where C is some constant. See graphs below.

## Bonus: Gyrobifastigium

The gyrobifastigium is a solid with eight faces -- four squares and four triangles. It can be thought of as putting two triangular prisms together along their square sides so that the triangles prisms are aligned perpendicularly. The gyrobifastigium is one of 92 shapes called Johnson solids, which are polyhedra composed of regular polygons that don't fit into the categories of Platonic solids, Archimedean solids, prisms, or antiprisms.

The name decomposes as gyro + bi + fastigium. The "fastigium" part comes from the Latin word for a sloping roof, as half the shape resembles the roof of a house. The "bi" part comes from the fact that there are two of them and the "gyro" part is because the two halves are rotated 90 degrees with respect to each other.

## How many of these weird shapes did you know?

**(Left)** An icosahedron is a polyhedron with 20 equilateral triangular faces where each vertex is the meeting point of five triangles.

**(Middle)** A Mobius strip is 2-dimensional surface with only one side.

**(Right)** A hendecagon is a polygon with 11 sides.