## The Trapezoid Trap

### What is this shape called?

The website that sells these “Nylon Trapezoid Multi-Color String Light Lanterns” indifferently claims to know. The suffix ‘-oid’, from Greek ‘-oeides’ means “of similar form to, but not the same as,” whatever precedes it, for example human/humanoid. Here are a few more examples:

- Asteroid (“like a star”)
- Thyroid (gland shaped “like a
*thyreos*or shield”) - Opioid (a compoud “like opium”)
- Steroid (a substance “like (chole)sterol”)
- Tabloid (“like a tablet”, i.e.
*tablet journalism*: small condensed articles) - Typhoid (“like typhus fever”)

The suffix is used oftentimes in mathematics too, for example:

- Cardioid (a shape “like a heart”)

- Paraboloid (a 3d surface “like a parabola”, created by rotating a parabola about its line of symmetry)

- Toroid (“like a torus”)

- Centroid (The center of mass, inertia, or gravity of a body or system of bodies)

- Ovoid (3d shape “like an oval” or egg)

- Cuboid (“like a cube”)

- Ellipsoid (3d shape “like an ellipse”)

- Spheroid (3d shape “like a sphere”; the Earth is actually an “oblate spheroid”, i.e not a perfect sphere due to its being slightly squished at the poles)

- And hundreds more! See here.

**So what about a trapezoid?** Well common sense would tell us that a trapezoid is so called because it’s shaped “like a trapeze”.

In America, a trapezoid is a quadrilateral with one pair of parallel sides.

In Britain, a trapezoid is a quadrilateral with *no *parallel sides.

(Conversely, in America a *trapezium *is a quadrilateral with *no *parallel sides,

and in Britain, a *trapezium *is a quadrilateral with *one pair* of parallel sides, i.e it’s the other way round.)

But a static trapeze, as used in circuses (circi?) since time immemorial, does have one pair of parallel sides (the roof and the horizontal rod), so the trapezoid is “like a trapeze”!! I’m afraid I’ll have to ‘side’ with the yanks on this one!

### So what’s the 3d trapezoid called?

Well it’s “like a trapezoid” so how about a *trapezoidoid*? But that’s ridiculous. It seems there’s no overriding systematic nomenclature for mathematical shapes. An ovoid is a 3d oval, so why isn’t a trapezoid a 3d trapeze(-ium), i.e the shape at the top of this post?

We need to look at it in a different way. It’s not a shape that has been rotated into the 3rd dimension (like an oval is to become an ovoid), so it’s more like a type of prism, i.e. a solid having bases or ends that are parallel. But those ends are not the same size.

It’s actually a *truncated square pyramid* (click the link – there’s a neat Java applet that allows you to click and play with the shape). That itself is what’s known as a “special case” of the *Pyramidal Frustum*, special due to its rotational and reflective symmetries.

I should mention that the inspiration for and content of this post resulted from a conversation I had with a old friend from when I studied applied maths at Nottingham University. In trying to nail down the correct name for that shape we came up with a few bizarre and interesting shapes, some of which are real. See if you can imagine what they look like!

- trapezoidal prism
- parabolapod
- cuboid with a shrunken face
- irregular quadrilateroid
- hexahedron
- hyperbolic trapezosphere
- prismatoid
- parallelapiped

I’ll write in earnest to oogalights.com asking them to change the name of this product to** ****“Nylon Truncated Square Pyramid Multi-Color String Light Lanterns”**. Or perhaps I won’t waste my time. They look terrible quality, and at $22.75 they’re a rip-off! I bet I could make one of them in less than 20 minutes for 50p.

the earth is not a sphere,ess or an ecllipes the earth is an a sperical shape in out side but in inside is an a flat only.

Keep us posting!

Thanks to Martin Nelson for the inspiration for this post