Tuesday, April 21, 2020

What is the Name for a 3-d Rectangle?

What do you think is the name for the generalization of a cube? Can you picture it? The shape has rectangular faces, and right angles at every vertex. It is the 3-d equivalent of a rectangle.

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This nameless yet ever-present "3-d rectangle" is the focus of this post. I would venture to say that it is the most often encountered geometric approximation in waking life. Planks, books, computers, and bricks all have this shape. It is the ultimate building block shape that most human structures are composed of. There are likely several physical approximations of this shape directly within your eyesight right now. Despite its incredible commonality, I would be surprised if anyone reading knows an indisputably correct geometric name for it off the top of their head.

The fact that there is no common word for this simple, common, and easily defined object is disappointing and confusing. And this doesn't just annoy me with my neurotic perspective on language, but it affects the children as well. I imagine my little sister or myself at a younger age, but there are countless geometrically curious 3rd graders who may innocently decide to google for the name of this shape, only to be rewarded with a frustrating spaghetti of language and math that even a three page essay cannot fully unravel.

Such a dose of linguistic absurdity for a curious 3rd grader may send them tumbling into their first, miniature existential crisis. It can be powerfully disillusioning, particularly to the very children for whom the precision and clarity of mathematics appeals. The terminology needs an update.


Box


The first and best guess many will take as a name for this shape is "box". Like cube, this word is concise and pleasant sounding. Unfortunately there are two major issues with this word. Firstly, in mathematical literature, box has been hijacked for a slightly different concept, a hypernym of the 3-d rectangle. Box is a synonym for hyperrectangle a rectangle generalized to any higher dimension. Thus, a 3-box would successfully correspond to our target object, but box alone as a mathematical term is too broad.

Orthographic Projection of a 6-Box with Regular Edges (6-Cube)

Despite not having exactly the right sense in the mathematical literature, 'box' still might be a fine word to represent the concept, and it is used this way by many. You could imagine that, since we are most familiar with 3-dimensional space, that box is a reasonable shortening of 3-box for ordinary use, in the same way that cube is a synonym for 3-cube. Unfortunately there is a subtle but important difference for the way mathematicians use box and cube. Cube originally and predominantly refers to the 3-dimensional variant, but box refers to the hypernym of the 3-box. Box alone is more of an analogous concept to hypercube, even though n-box and n-cube are analogous concepts.

That was probably confusing, but put simply: geometers do not ordinarily use box without a suffix to refer to a 3-dimensional box.

Still, it might be alright to extricate box from its prescribed definition in the mathematical literature and use it anyway in ordinary language, except for the second issue in the way box is used in the common vernacular rather than mathematical literature. This is perhaps the more important issue; box does not carry the connotations of abstractness and geometry. The use of box in common language is much more related to packaging than to any object in the abstract realm. The word is still firmly rooted in its etymological origin of ancient Greek, where it meant "wooden container". It is a word born from the concrete physical world that was halfheartedly adopted into geometry, not one that evokes the kind of abstract sense commanded by cube or circle.

Box

As a thought experiment to support this connotational inadequacy of box: conjure up a mental image of a box-shape, then make it as long and skinny as possible. Is it still box-shaped? In most people's intuition I would wager not it becomes more of a beam. Something long and skinny cannot be a box.

Beam


Also consider: is a brick a box-shaped object? Is the chair made up of several box-shaped segments? Perhaps I am equivocating now, but if so: the ease of equivocating with the word's meaning is one of the biggest problems with it. When we are dealing with entirely abstract geometric concepts, we want clear and closed definitions to represent them, and box does not do the trick, even though I wish it could.

Cuboid


The second guess many will have is cuboid. This makes some sense: the -oid suffix means "resembling", and our target shape is a cube-resembling shape. Although used frequently and successfully in common English, in some mathematical literature, cuboid means something subtly and confusingly different. Technically, the only constraints for a cuboid are that it has 6 faces, 8 vertices, and 12 edges. In an actual cuboid, none of the angles need to be 90 degrees, and none of the lines need to be parallel, meaning something like the following shape is considered cuboid.

Square Frustum: a Cuboid

Although cuboid, like box, is a hypernym of a 3-box in mathematical literature, in ordinary English it is the go-to word for the shape in question. Performing an internet search for most variations of the question: "What is the name of a 3-d rectangle?" will result in a forum post or dictionary definition about cuboid. It is unfortunate then that in geometric language this definition of cuboid is not agreed upon. However, cuboid can be particularized to our target shape by prefixing with "rectangular" or "right", as in "rectangular cuboid" or "right cuboid". Either of these modifiers add the requirement that all angles in the cuboid are 90 degrees. Then, have we solved the puzzle? Not really.

Right cuboid may be technically correct, but it is not commonly used at all. Counter-intuitively, people who already thought cuboid alone to be the correct word for the shape may assume that a cuboid being right is some deformation. Right cuboid may never become a widely understood term for a simple reason. It has the inelegant fate of being composed of two words. I believe that such a basic and common shape as the right cuboid deserves an equally basic and common single word name.

Rectangular Prism


One of the responses I have gotten most often when asking people for the name of this shape is rectangular prism. This is almost correct, but again is not quite specific enough. And, it shares the inelegance of right cuboid, being also composed of two words.


A prism is a shape with two polygons oriented across from one another. The polygons are congruent and parallel. A right prism is the more familiar kind, but prisms can be oblique as well when the bases are not lined up.

Using these terminological distinctions, the shape we are trying to describe is specifically a right rectangular prism. People usually aren't concerned with oblique prisms, so it seems fair to accept rectangular prism as a successful nickname at least, even if it is technically a hypernym. However, there is one other problem with the word. Like box, this problem is in the connotation.

Referring to the shape in question as a prism emphasizes the fact that the shape has two bases connected by rectangular sides. Ordinarily, this would allow us to coherently talk about height and base dimensions... But on this shape the bases are rectangles as well. How would you choose the bases? There are 3 separate ways of interpreting any given rectangular prism. If we want a word to wholly describe this shape, it should encompass all three interpretations at once rather than cause the eyes to continuously jump between different possibilities like an optical illusion.

Rectangular prism, like cuboid, does a decent job at picking out the shape we want. But we can do better.

Parallelepiped


Parallelepiped, another commonly used name for our shape, does not do any better. Parallelepiped technically refers to a class of shapes similar to a rectangular prism. The only constraint on a parallelepiped is that it has 6 faces that are all parallelograms. Our target shape is a kind of parallelepiped, but we could also imagine a parallelepiped where every face is a rhombus instead.

Parallelepiped

In order for a parallelepiped to become the shape we are interested in, it must have a rectangle for every face. This would be known as a rectangular parallelepiped. Once again, an inelegant complicated name for a simple thing. Of course, people still commonly and sucessfully use the more general word to pick out a rectangular parallelepiped, but other forms of actual parallelepipeds are not so rare themselves to be undeserving of a name in the common parlance.

Orthotope


Of existing words, Orthotope may be the best we can get.

Earlier I mentioned that box, aside from its non-geometric connotations, might work quite well as the word we are looking for. Fortunately, in geometry, box has a synonym, one that doesn't share any of its connotational issues.

Orthotope is derived from the roots ortho meaning right or straight, and tope often used in geometry as a suffix for shapes with arbitrary dimension. In the same way as with box, just "orthotope" can function as a suitable nickname for 3-orthotope since higher dimension objects don't normally enter into the conversation. This word works pretty well, having the kind of pure geometric connotations that Socrates would approve of, while still being composed of a single word that isn't overly technical sounding. The only gripe which remains is that the 'tope' suffix alludes to a more abstract shape than what we intend using this name for our shape would be like calling all cubes 'hypercubes'.

Orthopiped


Although orthotope is definitely a step in the right direction, I'd like to propose a new name: Orthopiped.

Orthopiped is a natural step from orthotope, following the analogue of parallelepiped for 3-dimensional and parallelotope for arbitrary dimension. The roots line up nicely as well: ortho meaning right, piped meaning surface/plane. It is the only closed shape that can be formed from any amount of surfaces all positioned at right angles.

Orthopiped := The 3-d closed shape formed by 6 flat faces where every vertex angle is a right angle.

The word is to the point, carries the geometric intuitions nicely, and isn't simply an adjective tacked on to the beginning of a more general shape.

Hierarchy of Shapes Discussed

Upgrading Geometric Terminology


Some might say that by making this article, I'm making a mountain out of a molehill, or worse, that I am wasting time in a futile effort to make mathematical terminology consistent.

It's true that mathematical terminology fails to reach its aspirations of being closed and consistent, and any attempts are likely futile. With this article I am not trying to "fix mathematical language" or anything of the sort. I just propose a word as a new option in the discourse, that in certain contexts can reduce confusion.

Next time you are answering a child's geometry question, or writing your mathematics paper that involves the shape discussed here, consider carefully your word choice and be explicit about defining it, whatever you choose. I think Orthopiped is your best option.

1 comment:

Anonymous said...

I just read this to the whole family and we thoroughly enjoyed it as an unexpected bedtime story.

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