I’ve had a few thoughts and discussions since I wrote this article about drawing stars. I thought it was worth sharing.
Stars within stars
The first observation is that stars, drawn in the way I described, contain other stars, nested within one another like a set of Russian dolls. Recall that we use the term ‘n-k star‘ to indicate a star with n points such that, if we draw a circle through all the points and consider the boundary of that circle as split up into n curve segments bounded by the points, then the straight line from one point to another traverses k of those curve segments. Like this, for a 7-3 star:
Yuriy made an interesting observation about the stars in my last article, that the sum of the angles of all the points of an n-k star is 180∘ × (n-2k). In the course of thinking of ways to prove that formula true, I came upon the realisation that a n-k star contains a n-(k-1)star, which contains a n-(k-2) star and so on down to the n-1 star, which is a regular, n-sided polygon.
Here’s a picture that shows this for an 11-5 star.
The red, outer 11-5 star contains a green 11-4 star, which contains a red 11-3 star, which contains a green 11-2 star, which contains a red 11-1 star (polygon). The points of each inner star are the innermost vertices of the star that immediately contains it. Since we will be referring to those vertices again later, let’s make up a name for them. We’ll call such a vertex a ‘tniop‘, since it is in a sense the opposite of a point. The above diagram shows a point and a tniop. We’ll call the stars inside a star ‘sub-stars‘.
We saw in my last essay that, when a star cannot be drawn without taking the pencil off the paper, it is made of a number of ‘component stars’, that are rotated copies of one another. Here is a picture of a 16-6 star, which uses different colours to highlight the two component stars. We have two 8-3 stars, one light blue and one red.
And here is a picture that uses colour variation to show the sub-stars of each of the two components.
The 8-3 light blue star contains an 8-2 star which is made up of two components that are 4-1 stars (also known as ‘squares’) coloured blue and pink. Similarly, the 8-3 red star contains an 8-2 star which is made up of two components that are 4-1 stars (also known as ‘squares’) coloured yellow and green.
The 8-2 stars each contain an 8-1 star (octagon) as the intersection of two squares -pink and dark blue for one octagon and yellow and green for the other.
Finally, those two octagons between them, bound a hexadecagon (16-sided polygon or 16-1 star). So altogether, in the one picture, we have:
- one 16-6 star (red and light blue)
- one 16-5 star (also red and light blue) inside that
- one 16-4 star (pink, green, dark blue and yellow) inside that
- one 16-3 star (also pink, green, dark blue and yellow) inside that
- one 16-2 star (also pink, green, dark blue and yellow) inside that
- one 16-1 star (also pink, green, dark blue and yellow) inside that
- two 8-3 stars (one red, one blue) making up the 16-6 star
- two 8-2 stars (one pink and dark blue, one yellow and green), one inside each of the 8-3 stars
- two 8-1 stars (octagon: one pink and dark blue, one yellow and green), one inside each of the 8-2 stars
- four 4-1 stars (squares: coloured pink, dark blue, yellow and green) which, in pairs, make up the 8-2 stars.
That’s sixteen stars altogether. What a lot of stars in one drawing! Can you see them all?
Here’s a different colouring that makes it easy to see all five 16-point stars:
Although the stars get smaller as k reduces, they do not shrink away to nearly nothing. In fact they get closer together as they go inwards, as if they are asymptotically approaching a circle of some fixed, minimum size.
To investigate this, I drew a 101-50 star:
You’ve probably noticed by now that I’m no longer drawing these by hand. My drawing is much too wobbly to capture the intricacies of stars-within-stars. So I wrote a computer program to draw them for me. I’ll try to remember to attach it at the end of the article, so that those of you who like mucking about with computers can muck about with it.
Anyway, that 101-50 star pretty well killed my hypothesis that the inner stars can’t get very small. In this one they almost disappear out of sight. I like the swirly patterns. I haven’t yet worked out whether they are really features of this very spiky, very complex, star, or whether they are just artefacts of the crudeness introduced by the computer’s need to pixillate.
Here’s a zoomed-in image of the interior of that star. Cool, eh?
This is a low-resolution image. I have saved a moderately high-resolution image of this star here. Zooming in and out is fun. It seems almost fractal as more patterns emerge from the inside when we zoom in. Also, the stars give the illusion that they are rotating as we zoom. To get the best effect you need to download the file (a .png image file) and then open it up, so that zooming is not limited by your internet connection’s speed.
Ratio of Outer to Inner radius
Let me briefly pick up on that idea above about whether there is some minimum inner radius for these stars. For each n, the spikiest n-k star is where k is the largest integer less than n/2, and this contains another k-1 stars, nested one within the other, down to the innermost, which is a n-sided regular polygon. We can work out the ratios of each star to the one immediately inside it, and use that to work out the ratio of the outermost, n-k, star, to the innermost, n-1, star. The attached computer program contains trigonometric formulas to do that. Here are the ratios of the radii of the innermost to the outermost star for each n from 1 to 109:
We observe that the ratios generally go down as n increases, but the decline is not steady. It bumps up and down. I have highlighted the prime numbers with asterisks. Notice how the ratio for those is always lower than for the numbers around them. The two drivers of the ratio seem to be:
- the size of n: the ratio generally declines as n increases; and
- the number of different factors n has. Note how 16 (divisible by 2, 4, 8) has a higher ratio than 15 (divisible by 3, 5) and 56 (divisible by 2, 4, 7, 8, 14) has a higher ratio than 55 (divisible by 5, 11).
This would be interesting to look into further, and to see if there is some neat, sweet, compact formula for the ratio that highlights the relationship to size and number of factors (if there really is one). But I have to stop thinking about that now or I’ll never post this essay.
The most general form of symmetrical stars
We can make an awful lot of stars using the above approach. For an integer n the number of different n-pointed stars is the biggest integer less than n/2.
But in fact there is an infinite number of different n-pointed stars, without having to loosen our standards by allowing asymmetry. After a bit of thought, I realised that the most general form of n-pointed star can be specified by a single number, which is the ratio of its inner radius to its outer radius. The outer radius is the radius of the circle on which the points sit. The inner radius is the radius of the circle on which all the tniops sit. Given n and that ratio – call it θ, we can draw a star as follows:
- Draw two concentric circles with ratio of the inner to the outer radius being θ.
- Mark n equally-spaced dots around the outer circle and draw faint lines connecting each of these to the centre. These will be the points of our star.
- Mark a dot on the inner circle halfway between each of the faint, radial lines drawn in the previous step. These dots will the the tniops of our star.
- Working consistently in one direction around the circle, draw a zig-zag line fro point to tniop to point to tniop and so on, always connecting to the nearest dot.
We will describe such a star as a n/θ star. We use a slash rather than a dash in order not to mix it up with the former type of star. Here is a sequence of six-point stars, with the ratio of the inner to outer radius going from 0.2 up to 0.8:
It is nice that this gives us more options for stars – infinitely many different kinds of star for each n in fact. But they are not as much fun to draw as the n-k stars, and it is harder to make them come out right without geometric instruments – which rules them out as an effective doodling pastime.
Note how, unlike with the n-k stars, the line leading away from a point does not intersect any other point. Nor do we get any inner stars for free. The price of gaining more variety is a loss of structure. The inner structure of a n-k star provides a great richness by enforcing all sorts of relationships between the vertices.
Only very specific values of the radius ratio θ give us n-k stars. I worked out the formula for the ratio by the way, with a bit of trigonometry. The ratio of the tniop radius to the point radius for a n-k star is:
2 sin(π(k-1)/n) sin(πk/n) sin(π(1-(2k-1)/n)) / (sin(2π/2)+sin(π(1-(2k-1)/n)) )
– sin(π((2k-1)/n – ½) )
For a n/θ star, if there are no integers n and k that give values of that formula equal to θ, the star will not have the array of inner stars that a n-k star has.
Computer Program to draw pretty stars
This whole diversion started as an exercise in drawing stars by hand. But there’s a limit to how intricate those stars can get without getting too messy with smeared ink or graphite. For those that like to look at pretty, intricate geometrical pictures, here’s my computer program in R that can draw stars of any of the types discussed.
Sums of angles
For those that like mathematical proofs, there are outlines of proofs here of Yuriy’s observation about the sum of the internal angles at points of a n-k star.
Bondi Junction, April 2017