The Secret Garden of Maths

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.