Volume of a Scutoid

Scutoid-mania

For many, maths was a difficult and unpleasant subject best left behind in memories of high school with acne and several embarrassing attempts at poetry. This is unfortunate as mathematics is not only all around us and part of everyday life (as enthusiastic maths communicators always insist to a sighing, eye-rolling crowd) but it is a creative and inventive field at its frontier. Unfortunately, so much time is allocated in math class to the mechanical and repetitive execution of monotonous calculations that very few people get a chance to see this.

As a result, it is very rare for mathematical discoveries to make headline news. This week has been an exception, with news sources around the globe announcing mathematicians and/or scientists have discovered a "new shape": the scutoid. There might be any number of reasons why this story is interesting to the public, though I think the fact that it can be communicated with pretty images and explained without numbers certainly helps. Of course, discovering a new shape is something of poor description for a few reasons (including that a shape is simply a descriptive concept and cannot really be "new", and that simply "discovering" a shape is a trivially simple task achieved by adding a twist or lump to an existing shape), but I don't see any reason to let that get in the way of an all-too-rare public fascination with geometry.

What is a scutoid?

The short answer is 'this'.

At it's most basic, it is a three-dimensional shape (or "polyhedra") with a top and a base that are parallel (usually shown being horizontal). At least one set of edges between the top and base forms a 'Y' shape, the others are straight lines. Wikipedia adds that "lateral faces are not necessarily planar" which is a fancy way of saying the sides don't have to be flat; they can curve which allows them to stack together nicely. This stacking property is important for reasons explained in the next paragraphs, but for this blog we will assume a scutoid has only flat sides for simplicity; the curvature could take a variety of shapes and variations that would be impossible to list and discuss exhaustively: regular curves like a circle, gradually changing curves like an ellipse or oval, 'S' shaped curves, etc. (For the pedants who note that faces with in excess of three vertices may not necessarily be flat as the vertices may not lie on the same plane, we will be assuming all such planes are divided into necessarily flat triangles with edges not shown in the diagrams).

So why are scutoids important? Mathematicians can invent any number of shapes, but we normally don't bother to give them unique names. Well, scutoids were developed to answer a question about epithelia--a type of tissue in animal bodies. Specifically, it was wondered how cells could pack together to form curved tissues since most of our organs don't form sharp angles. Flat shapes are easy to pack together; squares fit together like tiles on a floor, hexagons tesselate well in honeycomb, and many other regular and irregular shapes can be clipped together to create a solid, hole-less pattern.

Examples of flat tesselation. Clockwise from top left: honeycomb made of interlocking hexagons, a geometric pattern of perfectly fitting triangles, stylised lizards tesselating in the style of M. C. Escher, and square tiles.

Scutoids allow 3D interlocking to form flat planes or curved surfaces as required depending on the exact dimensions, as shown in this diagram in Nature. Because these shapes are important in cellular biology, it became convenient to give them a name and definition, leading to the "discovery" of scutoids and the recent media attention. Unfortunately, some people insist on bringing their anti-maths baggage to the party, though.

No doubt this sounds daunting to some people, but to me, this is a prime example of the enjoyment that can be found in math. Here we have a problem that may not ever have been tackled before. We can't just follow the mechanical, boring, unthinking steps set out in the math textbook and check the answer at the back. We need to actually do some problem solving and come up with a creative (though perhaps not unique) solution.

The volume of a scutoid

While most illustrations of scutoids in the media have been similar to those shown in my first image--pentagonal and hexagonal faces at the top and bottom joined with four linear edges and one 'Y' shape--a large variety of other forms are possible. The top and base can be triangles, squares or any other shape. These top-and-bottom faces do not need to have roughly equal-length sides. They can have reflex angles so the top could be zig-zag shaped, for example. The scutoid could have 2, 3 or more 'Y' shapes, and these could meet very close to one end, the other, or somewhere closer to the middle. Normally the centre of the 'Y' is shown vertically above one corner of the base, but it can be to the right or left and can protrude out from the shape or recede back into the scutoid.

This means there are a lot of variables to consider. I have also seen suggestions on how to approach this elsewhere online. Some people, probably considering the possible curves in the faces, have suggested using calculus while others advise calculating the volume of the shape 'before the triangle was removed' and then working out the volume of the 'missing' part and subtracting it. Both of these are great ideas, but I want to try something a little more accessible to the less mathematically inclined than calculus, and some forms of scutoid with a 'Y' fork far removed from the other points can have quite complex shapes before being truncated.

If you have never understood the enjoyment of maths firstly, it's frankly amazing you got this far into this post and secondly, I'd like to suggest you try and work out each step on your own as we go and see if you can find an answer before I spoil it. First, let's just play with a simple scutoid, before we get bogged down in the complicated variations. Let's use the pentagon-and-hexagon form commonly shown in the media, and easier yet let's use regular pentagons and hexagons, where all the sides and angles are equal.

Now there are several ways we could join the base to the top, but to keep this simple we will keep the bottom of the hexagon directly above the bottom pentagon, connect the top corner ("vertex") of the pentagon to the top two corners ("vertices") of the hexagon with a 'Y' and all the other corners to each other with lines.

And let's have the 'Y' fork half-way between the base and the top (though this actually won't matter, for reasons that will soon become obvious) and have the 'Y' flat rather than pyramid-like (i.e. the fork is halfway up an imaginary line that goes from the pentagon's vertex to the middle of the hexagon's side). That pretty well gives us a decent scutoid to be starting on.

Now we have defined our shape, we can start thinking about how to calculate its volume. Since the most complex part is the change in the shape's cross-section half-way up, my advice is to cut it in half so we have two shapes:

We will look at the bottom half first, as that seems simplest.

Volume below the fork

So this part almost looks like a prism, and prisms are pretty easy to calculate. A prism in geometry is basically a single straight-edged shape (or "polygon") stretched into the third dimension to make a kind of box; in fact, a square or rectangular prism is a regular everyday box, and a cube is a special kind of square prism. Because of this, the volume can be calculated easily by multiplying the area of the 2d face by the length the shape is stretched or 'extruded'.

All of the above are prisms, except the cylinder which has curved edges. However, because a cylinder is a circle extruded into the third dimension its volume can still be calculated by multiplying the area of the circle by the third dimension (in this case the height of the cylinder).

Unfortunately, our lower half-scutoid is not a perfect prism, as the pentagon at the top is not identical to that at the bottom. The edges do not rise in straight vertical lines as they move to line up with the vertices of the hexagonal face. The resulting shape is instead known as a "prismatoid" (specifically a type of "prismoid" called a "frustum") and can be somewhat less pleasant to calculate. Fortunately, we don't need to discover this bit ourselves, as there is already a known formula for the volume of a prismatoid:

V = h(T+4M+B)/6

This can look pretty intimidating, but that's normal for any formula before we understand what we are looking at. V is the volume we are looking for, h is the height of our prismatoid, T is the area of the top of the shape, B is the area of the base and M is the area half-way up. So the volume of our half-scutoid (Vlower) can be found by getting the area of the half-way-up cross-section, multiplying it by 4, adding the area of the top and the base, dividing the result by six and multiplying it by the height.

We can get into detailed maths of our particular scutoid, but we don't need to. We already have the formula we need so we can steal that and reuse it. As a fun exercise at home, we can easily calculate the area of the regular pentagonal base using Google, but can you determine the areas of T and M which are not regular pentagons? Here are some hints:

1. The edges of the scutoid move smoothly from the corners of the pentagon to the corners of the hexagon (or the middle of an edge in the case of the 'Y' edge). Since the shape was cut half-way up, the corners of T will be half-way between the pentagon's and the hexagon's, and the corners of M will be one-quarter of the way from the pentagon's.
2. A regular pentagon has angles of 108°. A regular hexagon has angles of 120°. Calculations for regular pentagons and hexagons can be assisted by the calculators at the other end of those links.
   

   

   Diagram of the scutoid (black), its edges (red), some useful dimensions taken from the linked calculators (purple) and the perimeter of the area T (green) to get you started.

3. To calculate the area of a pentagon, you can cut it into triangles and add them together. The lengths and angles of triangles can be calculated using a trigonometry calculator like this one. The area of a triangle is always equal to the length of its base x its height:

So we now know the bottom half of the scutoid has a volume of h(T+4M+B)/6. The letter T for top doesn't quite make sense in relation to the whole scutoid, so let's rename it F (the cross-section at the fork). The whole volume of the scutoid, then, = Vupper+h(F+4M+B)/6.

Volume above the fork

I stated that the area below the fork was a prismatoid without really explaining what a prismatoid is. According to Wikipedia
(and we all know you shouldn't trust the internet for important work, but we are all here for the pure fun of three-dimensional geometric fun, right? Right?) "a prismatoid is a polyhedron whose vertices all lie in two parallel planes. Its lateral faces can be trapezoids or triangles."

In non-nerd speak, then, a prismatoid is a 3D shape with flat faces and straight edges, with all its corners either at the bottom or the top where the bottom and top are parallel. (Our prisms fit this rule, the scutoid does not since it has a corner part way up as part of the 'Y'.) The 'sides' ("lateral faces") can be made with four sides and corners ("trapezoids", which include squares, rectangles and parallelograms) or three sides and corners (called "triangles, as we all know).

Bellow the fork, all of the corners are either at the pentagonal base or at the parallel cut, the faces are all flat, the edges are all straight and the "lateral faces" are all trapezoids. But interestingly, the shape above the fork is also a prismatoid too: all of the corners are either at the top hexagon or at the parallel cut, the faces are all flat, the edges are all straight and the "lateral faces" are all trapezoids except for one triangle above the fork. Our scutoid might not be a prismatoid, but it is made of two prismatoids on top of each other!

That means we can use the same equation as before:

V = h(T+4M+B)/6

Obviously, we need to rename M in the top half-scutoid to distinguish it from the M in the bottom half-scutoid, so we will call this N, and the height h, which we will call l for loftiness. B for base or bottom needs to change too but fortunately, we have already defined the area of the cross-section at the fork as F so we have a letter for it. The volume of the top half-scutoid, then, is V = l(T+4N+F)/6.

T is easy to calculate using the previous regular hexagon calculator. F you may have already calculated for the volume of the bottom half-scutoid. The cross-section of area N will be an irregular hexagon; the non-forked edges will be intersected half-way between their location at the cut and their location at the hexagon, while the new sixth side will have a length of 1/2, being half-way between 0 (at the fork) and 1 (at the top). Perhaps you might like to try and calculate the area of N.

The volume of a scutoid

We can now add the two halves of the scutoid back together, to get:

Vscutoid = Vupper + Vlower = l(T+4N+F)/6 + h(F+4M+B)/6

So far so good, but how does this work for other kinds of scutoid I hear you enthusiastically wondering. Well, let's consider some ways other scutoids might differ from the one we were playing with:

• Dimensions. Other scutoids could be taller, shorter, fatter or thinner, but as I left all the actual numbers for those playing along at home this doesn't really affect the calculation. Changes to height will be factored into the values l and h, changes in other dimensions will be accounted for in the area of the various cross-sections.
• 'Y' fork location. The fork point was arbitrarily set half-way between the top and base, but because h and l are handled separately in the equation one can have the lower half-scutoid taller or shorter without issue. The fork itself can not only be pushed up and down in this way but moving it left or right, depressing it into the scutoid or drawing it out are all permitted in the equation, as the resulting changes to the shape do not stop the scutoid being the combination of two prismatoids.
• Polygon shape. The regular polygons used here are purely for mathematical convenience, but since prismatoids can have their volumes calculated regardless of the polygons used this does not cause a problem for the equation. Triangles, trapezia and other shapes will work just as well, and the angles and lengths do not need to be equal at all.
• 'Y' fork count. All scutoids must have at least one fork, but I have not read any description that requires they have only one. scutoids with two, three or more 'Y' shapes are permissible. If these forks all occur at one level (that is, along one plane parallel to the base and top) the current equation works exactly as written. If these forks exist at different heights, the scutoid will need to be split into more prismatoids and the equation will become longer but essentially this just involves adding on more examples of the prismatoid volume formula.
• Curved faces. Much of the discussion of scutoids seems to consider it a variation of the prism-prismoid-prismatoid kind of polyhedra. In practice, the inclusion of curved and warped faces is essential to the biological application of these shapes, which allows non-polyhedral shapes to still be considered scutoids. These curved faces are not covered by the current equation, and more details of the curves involved would be needed before they can be accounted for. The nature of scutoids, however, is such that some four-cornered faces might not be able to lie flat even in the example used here. The assumption of flat faces is important to the equation but can be ensured if triangular faces are permitted.