Thinking outside the box

Negative polygons and other mathematical creations

Two people walk into a room. Five people walk out. How many people are in the room?

It was in the early hours of, would you believe it, Friday the 13th in July of 2007 when I was driving from Bordeaux to Orleans and feeling so sleep-deprived that I had one of those moments of strange inspiration. In order to keep awake on night drives, I would try and set myself a mathematical task to solve. With the family asleep and almost no one else on the road, there was little else to do. It must be something peculiar, I thought, and the more peculiar it was the more likely it would stop me from dozing off. How about throwing in some negative numbers into an area of maths where they are normally excluded? Geometry. Polygons.

The formula for working out the total internal angle $\theta$ of a polygon by knowing the number of sides $n$ is something every school girl and boy knows:

$$\theta = (n-2)180°.$$

They also know that $n$ is an integer and it cannot be smaller than 3. When $n$ does equal 3, they also know it is a triangle so in this case, the total internal angle is

$$\theta = (3-2)180 = 180°.$$

Kid’s play. I wondered: what if $n$ was negative three (for example)? That would put the negative cat among the positive pigeons! Forget for a moment that negative sides of a polygon are impossible – though there was a time when even negative numbers didn’t exist, nor fractions, nor irrationals, nor imaginaries. Let’s take the risk! Let’s just treat it as a piece of algebra. The equation can accept it so why not try? So $n = -3$. What internal angle do we get?

$$\theta = (-3-2)180 = -5 \times 180 = -900°.$$

There, it gave us an answer. But what does it mean? A polygon with three negative sides has an internal angle of $-900°$? As it has three sides (albeit negative ones), then perhaps it is some kind of triangle. An anti-triangle? A negative sided triangle? Negative distances are used when dealing with virtual images in optics, for example, so why not here? But where does the angle $-900°$ come in? How does it relate to this negative shape?

Equilateral triangle. Subtracting the $60°$ inside the vertex leaves $300°$ outside the vertex

In my mind (I was driving remember!) I mentally drew an equilateral triangle. I knew already that the angle inside it was $180°$, meaning that the internal angle at each vertex was $60°$. I wondered what the angle on the outside of each vertex was (not to be confused with the external or exterior angle, which is the angle turned when going round the vertex, which is $180 – \text{internal angle}$). Well, at a vertex, the total angle all round is, of course, $360°$, so subtracting the $60°$ inside the vertex leaves $300°$ outside the vertex. As there are three identical vertices in an equilateral triangle that makes a total angle outside the triangle of:

$$3 \times 300 = 900°.$$

Letting it sink in…. Hang on! Oh, this is interesting! This angle outside a normal triangle is numerically equal to this ‘internal’ angle of a ‘negative sided’ triangle! Let’s write this as a rule:

The angle outside a triangle is equal to the internal angle of a negative triangle but opposite in sign.

Does that mean that the area outside a normal triangle is our negative sided triangle? Or is it the triangular hole left behind? At this point, I was still not sure exactly what a negative triangle was but at least we seem to have made a breakthrough. Although we have only done it for an equilateral triangle, the same would happen for irregular triangles. (Prove it to yourself!)

Would the same happen for a four sided negative polygon, i.e. an anti-quadrilateral? Let’s see. Using the same formula and setting $n$ to negative four, the total ‘internal’ angle is

$$(-4-2)180 = -6 \times 180 = -1080°.$$

If we take a regular quadrilateral, a square, it is easier to analyse. The total internal angle would be ($n = 4$)

$$(4-2)180 = 360°.$$

On the left, a square, so there’s $90°$ inside, and on the right, the $(360-90) = 270°$ outside each vertex.

This isn’t a surprise and nor is the internal angle at each vertex ($360/4$) being $90°$. What we want to know is the angle outside each vertex. With the total at a vertex being $360°$ and there’s $90°$ inside, that makes $(360-90) = 270°$ outside each vertex. Being four vertices makes a total angle outside as $4 \times 270$ which is $1080°$. The same numerical total we got earlier but the opposite sign. So a negative sided square (or in general any negative sided quadrilateral-prove it to yourself!) has a total ‘internal’ angle of $-1080°$.

You can prove to yourself that the same would happen for any negative sided polygon and thus we can write a general rule:

The total internal angle of a negative sided polygon is numerically equal to the total external angle of the equivalent positive sided polygon but negative in sign.

We still don’t fully know what a negative sided polygon is nor how it differs from the positive version (apart from this negative internal angle), so how about putting forward a hypothesis.

Law of conservation of edges

Well perhaps it should be called a conjecture, but I shall leave that decision to the experts. Anyway here it is:

The total number of edges is constant.

Okay, imagine a sheet of paper, which will represent a plane and could be infinite but we will draw it as being rectangular in shape so we are only looking at part of it. There are no edges, so the total number of edges is zero. Now we shall cut an equilateral triangle out of the paper and remove it (see figure below). I think we can agree that the cut out triangle has three edges. Three positive edges. We have created three positive edges (calling an edge $E$)

$$+E + E + E=+3E.$$

Removed equilateral triangle

Using the electrical convention (red positive, black negative), the edges have been coloured red in the diagram. Now if we invoke the conservation ‘rule’ above, the total number of edges must still be zero so there must be three negative edges created to balance the three positive ones

$$-E – E – E = -3E$$

so

$$+3E – 3E = 0.$$

Well, there they are, look, marking the edges of the triangular hole left in the paper! The triangular hole is the negative sided triangle. The absence of a triangle is the negative triangle! The anti-triangle. We have identified it!

Do we have an example of such a negative triangle in the real world? Well as it happens, yes! That kiddies’ toy with pegs and holes. The toy is usually a wooden base having different shaped holes cut into it: a circle, a triangle, and a square, and the child is supplied with wooden triangular, square and circular prisms and asked to fit them into the correct holes – see figure below. So it seems our negative sided polygons have been staring us in the face since we were kids!

If we create a few more similar rules:

the total number of faces is constant

and

the total number of vertices is constant

Hole left behind by the removed triangle

then surely we can also say that our cut out triangle also has one positive area and three positive vertices. In order to conserve area and vertex number, the triangular hole must have a negative area and three negative vertices. See figure below. Suppose the area of the triangle was $4m^{2}$. If this was removed from a sheet of paper then our universe is now smaller in area by $4m^{2}$, so the triangular hole must have an area of $-4m^{2}$ in order to conserve the quantity. Or using $+A$ for the area of the triangle and $–A$ for the negative area of the negative triangle then

$$-A + A = 0$$

and if the vertices are similarly labelled $+V$ and $–V$ then

$$-3V + 3V = 0.$$

These rules can be applied to all polygons with any number (positive or negative) of sides. Well almost.

Polygons with $|n| = 2$

So far I have avoided them, but there they are, taunting us from just below our comfort zone and calling out “what about us?!” Although three is considered to be the smallest number of sides for a polygon, it is possible to have a two-sided shape in non-Euclidean geometry, such as on a spherical surface. One can draw a line from one pole to the other on a sphere, then a second line starting and ending at the same points (the poles) but with a fixed angle between them at both poles. This is equivalent to drawing two lines of longitude on the earth at, say, the Greenwich Meridian and 30 degrees east. Such a two-sided polygon is called a digon.

Two sided polygon on a sphere

Let us suppose we could force a digon into our flat Euclidean space of a plane, then gradually squash the digon to make the two sides as straight as we can get them (see figure below). As the two sides become more parallel, the two internal angles approach zero, so the total internal angle of a digon would be zero. Using the polygon formula, when $n = 2$ we get

$$(2 – 2) 180 = 0 \times 180 = 0°.$$

Digon forced into the plane

So our illustration of a flat digon satisfies the formula in having no internal angle. Now we can consider a polygon with negative two sides. A negative digon. An anti-digon. Here $n = -2$ so the internal angle of such a polygon would be

$$(-2 -2)180 = -4 \times 180 = -720°.$$

Looking at the outside angle of the digon in the diagram above right, each vertex has no angle inside so the outside angle for each vertex must be $360°$ and as there are two vertices, the total outside angle is

$$2 \times 360 = 720°.$$

This result is consistent with all the other polygons with a negative integer number of sides. Is there a real life physical example of a negative sided digon? Well yes, there is! It is none other than the ordinary every day button hole!

A buttonhole is a negative sided digon!!

Thinking clockwise and anti-clockwise

In physics and maths, direction is important for vector quantities. If one direction is positive then the opposite direction is taken as negative. Up is positive so down is negative. The same is applied to rotation. As mathematicians take anticlockwise as the positive direction then we shall stick to that convention. One could now look at these internal and outside angles in terms of the direction of rotation. Take our tried and trusted equilateral triangle again and select one side (see figure below). If one measures the internal angle at this vertex by going anticlockwise it is 60°. As it is clockwise, then the angle is taken as positive, $+60°$. Starting at the same edge as before and going clockwise, one encounters the other side of the angle after 300°. As we have gone clockwise the angle is negative and hence $-300°$.

Clockwise and anticlockwise angles

This idea can be extended further to include not only the angles but the sides themselves. One could consider walking anti-clockwise around the perimeter of a triangle (see figure above) and due to the positive direction, the sides themselves would be positive. If one walked clockwise around the same triangle, the direction is negative, so the sides would be negative. This is treating the sides as vectors, with positives and negatives entering the field of play.

Such a consideration reminded me of an example from the topic of thermodynamics and heat cycles on pressure-volume graphs. Consider the reversible change shown in the figure below. (I won’t bore you with the details!) Here the gas does work on the surroundings and on this type of graph the area inside the cycle is the work done and it is said to be positive. Now if one was to go around the heat cycle in the other direction, work is not done by the gas on the surroundings, but on the gas by the surroundings and that is said to be negative.

Work done by a gas or on a gas

This is actually the opposite of the outcome with the triangles above because clockwise is taken as positive on $pV$ graphs, but this still illustrates the idea of positive and negative areas depending on direction chosen around the shape.

And this is only scratching the surface of the topic. It is like an iceberg, with so much waiting for us below. With fractional sided polygons already known (the pentagram for example with only $5/2$ sides), there are ways to show such fractional polygons continue below two sides and into the negative camp, as well as what I called overturned polygons (those with external angles greater than $180°$), and the biggest shock of all: polygons with a mixture of positive and negative sides! Who knows, one might even find some with imaginary sides…

To negative infinity and beyond

1. Flickr user Rich Anderson, cropped, CC BY-NC-SA 2.0; 2. Flickr user Nedra, cropped, CC BY-NC-SA 2.0; 3. Flickr user cc511 CC BY-NC-SA 2.0; 4. other pictures by Hugh Duncan

Hugh Duncan graduated from UCL in 1980 having studied astronomy. He teaches physics and maths in the International School of Nice.
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