The Geometry of Stars

The Geometry of Stars © 2013 Tofique Fatehi Page 1

T H E G E O M E T R Y O F S T A R S

May 10, 2013 Tofique Fatehi [email protected]

http://tofique.fatehi.us

PRELUDE

“Why ask for the moon, when we have the stars?” - Tony Brent

In this essay I try to categorize star-shaped Figures, so that they can be systematically classified and documented. Not much work seems to have been done in this field, and so, I take full liberty to venture into it, using my own descriptions and evolving my own terminology, where none are clearly available.

STARS

Just as a plane figure of “n” sides is called a polygon, we may term a plane n-pointed star a “polygram”. This term is not in general use, and may even have a completely different meaning in some other branch of science. Depending on the value of “n”, many of these figures have specific names. Polygons are named pentagon, hexagon and so on, and polygrams are named pentagram, hexagram and so on. These figures are said to be regular if all sides and all corresponding angles are equal.

Acknowledgment This is to acknowledge with thanks the meticulous effort put in by Parul Vijay Patil in the preparation of all the drawings for this essay.


Figure 1a shows a regular hexagon (a six-sided polygon). If all the sides of the hexagon are extended on both sides, it produces a six-pointed star (as in Figure 1b). This star is called a hexagram.

Figure 1a - Hexagon

Figure 1b - Hexagram

Figure 2a shows a regular pentagon (a five-sided polygon). This produces a five-pointed star if all the sides are extended on both sides (as in Figure 2b). This star is called a pentagram. It is also sometimes called a pentalpha or a pentangle.

Figure 2a – Pentagon

Figure 2b - Pentagram


The pentagram can be drawn fully, without lifting the pen from the paper (unicursally), and without breaking any side. The hexagram cannot be so drawn - at least not without breaking any side. In fact, the hexagram actually is composed of two interlaced equilateral triangles. Stars which can be drawn unicursally may be termed “prime”. Stars which are composed of interlaced figures may be termed “composite”.

Figure 4 shows a square and a triangle (four-sided and three-sided polygons). For these figures, even if the sides are extended indefinitely, no star is produced. It may therefore be concluded that the minimum number of points required to form a star is five.

Figure 4a – Square

Figure 4b – Triangle

Now let's take a look at a regular septagon (a seven-sided polygon). This is also called a heptagon. This is shown in Figure 5a. As before, extending all the sides both ways produces a seven-pointed

It may be pointed out that there does exist a unicursal hexagram (much used in witchcraft), but then, that does not belong to any class of our present study. Just for the record, this is shown in Figure 3.

Figure 3


star, called a septagram or heptagram, shown in Figure 5b. It does not end here. If the sides are extended still further, another seven-pointed star is produced (as in Figure 5c). This also has the same name: septagram (or heptagram).

Figure 5a - Septagon

Figure 5b -Septagram of the First Order

Figure 5c -Septagram of the Second Order

So now, we have two different seven-pointed stars, both regular, and both prime, and yet both different. We need to differentiate between these two forms. We do so by inventing an “order”. We shall call the first star (in Figure 5b) a “septagram of the first order”, and the second star (in Figure 5c) a “septagram of the second order”. It may be stated that there is no second order pentagram or hexagram. It is worth noting that any polygon can be considered a zero-order polygram.


An unspecified order of any star may be designated as “m”. At this point, we may introduce a notation to specify any star. We will use the notation “*n^m” to specify an “n” pointed star of order “m”. This will become clearer as we proceed.

There is a simpler way to draw the polygrams in a more manageable manner.

SIMPLIFYING CONSTRUCTION OF STARS

Suppose our target is an n-sided figure. First draw a circle (of any diameter, that's immaterial). Mark “n” points on the circumference, all equidistant. Number these points serially from 1 to “n”, (clockwise or otherwise, that's immaterial). If all these points (starting from point 1) are connected serially by a line, we end up returning to the starting point, and we get an “n”-sided regular polygon.

If we connect these points by a line, not to next point, but, to the next-but-one point, (that is, every alternate point) we will return back to the starting point. If we have covered all the points, we would have got an n-pointed prime polygram. However, it may so happen that we return to the starting point, without yet covering all the points. This will be an indication that the polygram is not prime, but composite, consisting of two (or more) interlaced figures, of which we have one figure.

Next, we start from a free point adjacent to the starting point of the first figure, and complete the circuit, following the same rule (of connecting the next-but-one point). If there are still any uncovered points, we repeat the process until all “n” points are covered. We would then have got an n-pointed composite polygram. This star, prime or composite, would be a first-order star.

Figure 6 – Construction of a Prime Polygram

Figure 7 – Construction of a Composite

Polygram


If the points are connected by a line to the next-but-two point, we get a prime star if we return to the starting point after covering all the points. Otherwise, if some of the points remain uncovered, then, following a procedure similar to the one in the previous paragraph, we get a composite star. Either prime or composite, this would be a second-order star.

All the higher ordered polygrams can also be similarly defined.

SIGNATURES

This sequence of connecting the points may be termed the “signature” of the polygon or polygram. It may be noted that each and every figure has its own unique signature.

Thus, the signature of a pentagon is (1 2 3 4 5 1), and the signature of a pentagram is (1 3 5 2 4 1).

The signatures of all polygons are self-evident (sequential, from 1 to “n” and back to 1) and so need no further consideration. Those of the polygrams are not so obvious, and hence are dealt with further.

The signature of the hexagram is (1 3 5 1) (2 4 6 2). Notice that the hexagram is a composite polygram consisting of two interlaced triangles. Hence we have the two groups of bracketed figures. This is the only star which has no prime figure.

Figure 6 –Prime pentagram *5^1,

with signature (1 3 5 2 4 1)

Figure 7 –Composite hexagram *6^1,

with signature (1 3 5 1) (2 4 6 2)


There are two septagrams, both prime. One is of the first order and the other is of the second order. The signature of the first order septagram is (1 3 5 7 2 4 6 1). The signature of the second order septagram is (1 4 7 3 6 2 5 1).

Figure 8 – Septagram *7^1,

with signature (1 3 5 7 2 4 6 1)

Figure 9 – Septagram *7^2,

with signature (1 4 7 3 6 2 5 1)

There are two octagrams. The first-order octagram is composite, being composed of two interlaced squares, and the second-order octagram is prime.

Figure 10 – Octagram *8^1,

with signature (1 3 5 7 1) (2 4 6 8 2)

Figure 11 - Octagram *8^2,

with signature (1 4 7 2 5 8 3 6 1)


A nine-pointed star is called a nonagram (or enneagram), and has three configurations. The first and third order stars are prime and the second order star is composite and is composed of three equilateral triangles.

Figure 12 - Nonagram *9^1,

with signature (1 3 5 7 9 2 4 6 8 1)


with signature (1 4 7 1) (2 5 8 2) (3 6 9 3)


with signature (1 5 9 4 8 3 7 2 6 1)


A ten-pointed star, called a decagram, has three configurations. The first-order star is composite, being composed of two interlaced pentagons. The second-order star is prime. The third-order decagram is composite. All previous composites have been composed of two or more polygons. This is the first instance when the composite star is composed of two interlaced pentagrams. When “n” is equal to ten, there are five even and five odd points, enabling us to have two interlaced pentagrams.

Figure 15 - Decagram *10^1,

with signature (1 3 5 7 9 1) (2 4 6 8 10 2)


with signature (1 4 7 10 3 6 9 2 5 8 1)


with signature (1 5 9 3 7 1) (2 6 10 4 8 2)


The eleven-pointed star, called hendecagram has four all prime configurations.

Figure 18 - Hendecagram *11^1,

with signature (1 3 5 7 9 11 2 4 6 8 10 1)


with signature (1 4 7 10 2 5 8 11 3 6 9 1)


with signature (1 5 9 2 6 10 3 7 11 4 8 1)


with signature (1 6 11 5 10 4 9 3 8 2 7 1)


The twelve-pointed star, called dodecagram, also has four configurations, where only the fourth-order star is prime. The first-order star is composed of two interlaced hexagons, the second-order star is composed of three interlaced squares, and the third-order star has four interlaced equilateral triangles as their compositions.

Figure 22 - Dodecagram *12^1,

with signature (1 3 5 7 9 11 1) (2 4 6 8 10 12 2)


with signature (1 4 7 10 1) (2 5 8 11 2) (3 6 9 12 3)


with signature (1 5 9 1) (2 6 10 2) (3 7 11 3) (4 8 12 4)


with signature (1 6 11 4 9 2 7 12 5 10 3 8 1)


The thirteen-pointed star, called tridecagram has five all prime configurations.

Figure 26 - Tridecagram *13^1,

with signature (1 3 5 7 9 11 13 2 4 6 8 10 12 1)


with signature (1 4 7 10 13 3 6 9 12 2 5 8 11 1)


with signature (1 5 9 13 4 8 12 3 7 11 2 6 10 1)


with signature (1 6 11 3 8 13 5 10 2 7 12 4 9 1)



with signature (1 7 13 6 12 5 11 4 10 3 9 2 8 1)

A fourteen-pointed star is called a tetradecagram. It has five configurations. The second and fourth order stars are prime. The first order star is composed of two interlaced septagons. The third and fifth order stars are composed of two interlaced septagrams: septagrams of the first order in one and septagrams of the second order in the other.

Figure 31 - Tetradecagram *14^1,

with signature (1 3 5 7 9 11 13 1) (2 4 6 8 10 12 14 2)


with signature (1 4 7 10 13 2 5 8 11 14 3 6 9 12 1)



with signature (1 5 9 13 3 7 11 1) (2 6 10 14 4 8 12 2)


with signature (1 6 11 2 7 12 3 8 13 4 9 14 5 10 1)


with signature (1 7 13 5 11 3 9 1) (2 8 14 6 12 4 10 2)


TABLE OF POLYGRAM SIGNATURES

Points “n” Order “m” Type Signature

5 1 Prime (1 3 5 2 4 1)

6 1 Composite (1 3 5 1) (2 4 6 2)

7 1 Prime (1 3 5 7 2 4 6 1)

7 2 Prime (1 4 7 3 6 2 5 1)

8 1 Composite (1 3 5 7 1) (2 4 6 8 2)

8 2 Prime (1 4 7 2 5 8 3 6 1)

9 1 Prime (1 3 5 7 9 2 4 6 8 1)

9 2 Composite (1 4 7 1) (2 5 8 2) (3 6 9 3)

9 3 Prime (1 5 9 4 8 3 7 2 6 1)

10 1 Composite (1 3 5 7 9 1) (2 4 6 8 10 2)

10 2 Prime (1 4 7 10 3 6 9 2 5 8 1)

10 3 Composite (1 5 9 3 7 1) (2 6 10 4 8 2)

11 1 Prime (1 3 5 7 9 11 2 4 6 8 10 1)

11 2 Prime (1 4 7 10 2 5 8 11 3 6 9 1)

11 3 Prime (1 5 9 2 6 10 3 7 11 4 8 1)

11 4 Prime (1 6 11 5 10 4 9 3 8 2 7 1)

12 1 Composite (1 3 5 7 9 11 1) (2 4 6 8 10 12 2)

12 2 Composite (1 4 7 10 1) (2 5 8 11 2) (3 6 9 12 3)

12 3 Composite (1 5 9 1) (2 6 10 2) (3 7 11 3) (4 8 12 4)

12 4 Prime (1 6 11 4 9 2 7 12 5 10 3 8 1)

13 1 Prime (1 3 5 7 9 11 13 2 4 6 8 10 12 1)

13 2 Prime (1 4 7 10 13 3 6 9 12 2 5 8 11 1)

13 3 Prime (1 5 9 13 4 8 12 3 7 11 2 6 10 1)

13 4 Prime (1 6 11 3 8 13 5 10 2 7 12 4 9 1)

13 5 Prime (1 7 13 6 12 5 11 4 10 3 9 2 8 1)

14 1 Composite (1 3 5 7 9 11 13 1) (2 4 6 8 10 12 14 2)

14 2 Prime (1 4 7 10 13 2 5 8 11 14 3 6 9 12 1)

14 3 Composite (1 5 9 13 3 7 11 1) (2 6 10 14 4 8 12 2)

14 4 Prime (1 6 11 2 7 12 3 8 13 4 9 14 5 10 1)

14 5 Composite (1 7 13 5 11 3 9 1) (2 8 14 6 12 4 10 2)


SOME OBSERVATIONS

These observations are given without any proofs. In most cases the proofs are self-evident, or easily derived.

In this class of stars, the first thing to notice is that at the core of each polygram is a polygon having the same number of sides as the polygram has points.

The minimum number of points a polygram can have is five which gives us a pentagram. In other words, the smallest value for “n” can be five.

For a given “n” there is a limit to the maximum value of “m”, that is, the highest order the polygram can have. This value is also the number of configurations the star can have. We will designate “M” to be the highest value “m” can have for a given “n”.

The value of “M” is given by the equation M = (n - 3)/ 2 after discarding any fractional part. If “n” is a prime number, then all of the “M” configurations of the star are prime polygrams. If “n” is not a prime number, then the prime factors of “n” will determine which of the “M” configurations are prime, and which are composite. Further, the prime factors of “n” will also determine the compositions of the composite polygrams.

A BRIEF INTRODUCTION TO HYBRID POLYGRAMS

Consider the two fourteen-pointed stars *14^3 and *14^5. Both these stars are composite stars composed of two septagrams. Whereas *14^3 is composed of two septagrams of the first order, the other star *14^5 is composed of two septagrams of the second order. But in both these stars, all the odd numbered points form one septagram, and all the even numbered points form the second septagram, which interlace each other.

Notice that the prime factors of 14 are 2 and 7 and for “n” equal to seven, “M” is equal to two.

Consider now, this new, and different fourteen-pointed star. Inscribe a septagram of the first order using all the odd numbered points and interlace it with another septagram of the second order inscribed in all the even numbered joints. Its signature is (1 5 9 13 3 7 11 1) (2 8 14 6 12 4 10 2).


Figure 36 - Hybrid Tetradecagram,

with signature (1 5 9 13 3 7 11 1) (2 8 14 6 12 4 10 2)

This is of a different class of stars. It is not regular, as all sides and all angles are not equal. The central core is not a fourteen-sided polygon, even though the star is a fourteen-pointed polygram. And yet, it is composed of two regular septagrams. This class of stars may be classified as “hybrid polygrams”.

Probably the best way to represent this hybrid tetradecagram is *(2x7)^(1, 2) indicating that there are two seven-pointed stars of the first and second orders. Or, in a more generalized form, *(k x n)^(m1 ,m2, m3, …, mk), indicating that there are “k” number of “m”-pointed stars of the several orders m1 ,m2, m3, …, mk.

We shall dwell bit more on this before calling it a day.

The next prime number after seven is eleven, and an eleven-pointed star (hendecagram) has four prime configurations. So, we can have numerous stars with:

2 x 11 = 22 points, composed of any two hendecagrams of different permissible orders, or

3 x 11 = 33 points, composed of any three hendecagrams of different permissible orders, or

4 x 11 = 44 points, composed of the four hendecagrams of different permissible orders

And then to the next prime, and the next … and the next, ad infinitum.


But that's not all folks. There are many more classes of stars than the two essayed here. So, as Perry Como sings, “don't let the stars get in your eyes”.

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The Geometry of Stars