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The Geometry of Stars

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The Geometry of Stars

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The Geometry of Stars

  1. 1. The Geometry of Stars© 2013 Tofique FatehiPage 1THE GEOMETRY OF STARSMay 10, 2013 Tofique Fatehitofiquef@yahoo.comhttp://tofique.fatehi.usPRELUDE“Why ask for the moon, when we have the stars?”- Tony BrentIn this essay I try to categorize star-shaped Figures, so thatthey can be systematically classified and documented. Notmuch work seems to have been done in this field, and so, Itake full liberty to venture into it, using my own descriptionsand evolving my own terminology, where none are clearlyavailable.STARSJust 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 insome other branch of science. Depending on the value of “n”, many of these figures have specificnames. 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 anglesare equal.AcknowledgmentThis is to acknowledgewith thanks themeticulous effort put in byParul Vijay Patil in thepreparation of all thedrawings for this essay.
  2. 2. The Geometry of Stars© 2013 Tofique FatehiPage 2Figure 1a shows a regular hexagon (a six-sided polygon). If all the sides of the hexagon areextended on both sides, it produces a six-pointed star (as in Figure 1b). This star is called ahexagram.Figure 1a - Hexagon Figure 1b - HexagramFigure 2a shows a regular pentagon (a five-sided polygon). This produces a five-pointed star if allthe sides are extended on both sides (as in Figure 2b). This star is called a pentagram. It is alsosometimes called a pentalpha or a pentangle.Figure 2a – Pentagon Figure 2b - Pentagram
  3. 3. The Geometry of Stars© 2013 Tofique FatehiPage 3The pentagram can be drawn fully, without lifting the pen from the paper (unicursally), andwithout breaking any side. The hexagram cannot be so drawn - at least not without breaking anyside. In fact, the hexagram actually is composed of two interlaced equilateral triangles. Starswhich can be drawn unicursally may be termed “prime”. Stars which are composed of interlacedfigures 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 thatthe minimum number of points required to form a star is five.Figure 4a – Square Figure 4b – TriangleNow lets 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-pointedIt may be pointed out that theredoes exist a unicursal hexagram(much used in witchcraft), butthen, that does not belong to anyclass of our present study. Justfor the record, this is shown inFigure 3.Figure 3
  4. 4. The Geometry of Stars© 2013 Tofique FatehiPage 4star, called a septagram or heptagram, shown in Figure 5b. It does not end here. If the sides areextended still further, another seven-pointed star is produced (as in Figure 5c). This also has thesame name: septagram (or heptagram).Figure 5a - Septagon Figure 5b -Septagram of the First OrderFigure 5c -Septagram of the Second OrderSo now, we have two different seven-pointed stars, both regular, and both prime, and yet bothdifferent. 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 (inFigure 5c) a “septagram of the second order”. It may be stated that there is no second orderpentagram or hexagram. It is worth noting that any polygon can be considered a zero-orderpolygram.
  5. 5. The Geometry of Stars© 2013 Tofique FatehiPage 5An unspecified order of any star may be designated as “m”. At this point, we may introduce anotation to specify any star. We will use the notation “*n^m” to specify an “n” pointed star oforder “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 STARSSuppose our target is an n-sided figure. First draw a circle (of any diameter, thats immaterial).Mark “n” points on the circumference, all equidistant. Number these points serially from 1 to “n”,(clockwise or otherwise, thats immaterial). If all these points (starting from point 1) areconnected serially by a line, we end up returning to the starting point, and we get an “n”-sidedregular polygon.If we connect these points by a line, not to next point, but, to the next-but-one point, (that is, everyalternate point) we will return back to the starting point. If we have covered all the points, wewould have got an n-pointed prime polygram. However, it may so happen that we return to thestarting point, without yet covering all the points. This will be an indication that the polygram isnot prime, but composite, consisting of two (or more) interlaced figures, of which we have onefigure.Next, we start from a free point adjacent to the starting point of the first figure, and complete thecircuit, following the same rule (of connecting the next-but-one point). If there are still anyuncovered points, we repeat the process until all “n” points are covered. We would then have gotan 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 CompositePolygram
  6. 6. The Geometry of Stars© 2013 Tofique FatehiPage 6If the points are connected by a line to the next-but-two point, we get a prime star if we return tothe 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.SIGNATURESThis sequence of connecting the points may be termed the “signature” of the polygon orpolygram. 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 soneed no further consideration. Those of the polygrams are not so obvious, and hence are dealtwith further.The signature of the hexagram is (1 3 5 1) (2 4 6 2). Notice that the hexagram is a compositepolygram consisting of two interlaced triangles. Hence we have the two groups of bracketedfigures. 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)
  7. 7. The Geometry of Stars© 2013 Tofique FatehiPage 7There are two septagrams, both prime. One is of the first order and the other is of the secondorder. The signature of the first order septagram is (1 3 5 7 2 4 6 1). The signature of the secondorder 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 twointerlaced 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)
  8. 8. The Geometry of Stars© 2013 Tofique FatehiPage 8A nine-pointed star is called a nonagram (or enneagram), and has three configurations. The firstand third order stars are prime and the second order star is composite and is composed of threeequilateral triangles.Figure 12 - Nonagram *9^1,with signature (1 3 5 7 9 2 4 6 8 1)Figure 13 - Nonagram *9^2,with signature (1 4 7 1) (2 5 8 2) (3 6 9 3)Figure 14 - Nonagram *9^3,with signature (1 5 9 4 8 3 7 2 6 1)
  9. 9. The Geometry of Stars© 2013 Tofique FatehiPage 9A 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-orderdecagram 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 twointerlaced pentagrams.Figure 15 - Decagram *10^1,with signature (1 3 5 7 9 1) (2 4 6 8 10 2)Figure 16 - Decagram *10^2,with signature (1 4 7 10 3 6 9 2 5 8 1)Figure 17 - Decagram *10^3,with signature (1 5 9 3 7 1) (2 6 10 4 8 2)
  10. 10. The Geometry of Stars© 2013 Tofique FatehiPage 10The 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)Figure 19 - Hendecagram *11^2,with signature (1 4 7 10 2 5 8 11 3 6 9 1)Figure 20 - Hendecagram *11^3,with signature (1 5 9 2 6 10 3 7 11 4 8 1)Figure 21 - Hendecagram *11^4,with signature (1 6 11 5 10 4 9 3 8 2 7 1)
  11. 11. The Geometry of Stars© 2013 Tofique FatehiPage 11The 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-orderstar is composed of three interlaced squares, and the third-order star has four interlacedequilateral 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)Figure 23 - Dodecagram *12^2,with signature(1 4 7 10 1) (2 5 8 11 2) (3 6 9 12 3)Figure 24 - Dodecagram *12^3,with signature(1 5 9 1) (2 6 10 2) (3 7 11 3) (4 8 12 4)Figure 25 - Dodecagram *12^4,with signature(1 6 11 4 9 2 7 12 5 10 3 8 1)
  12. 12. The Geometry of Stars© 2013 Tofique FatehiPage 12The 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)Figure 27 - Tridecagram *13^2,with signature (1 4 7 10 13 3 6 9 12 2 5 8 11 1)Figure 28 - Tridecagram *13^3,with signature(1 5 9 13 4 8 12 3 7 11 2 6 10 1)Figure 29 - Tridecagram *13^4,with signature(1 6 11 3 8 13 5 10 2 7 12 4 9 1)
  13. 13. The Geometry of Stars© 2013 Tofique FatehiPage 13Figure 20 - Tridecagram *13^5,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 andfourth order stars are prime. The first order star is composed of two interlaced septagons. Thethird and fifth order stars are composed of two interlaced septagrams: septagrams of the firstorder 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)Figure 32 - Tetradecagram *14^2,with signature(1 4 7 10 13 2 5 8 11 14 3 6 9 12 1)
  14. 14. The Geometry of Stars© 2013 Tofique FatehiPage 14Figure 33 - Tetradecagram *14^3,with signature(1 5 9 13 3 7 11 1) (2 6 10 14 4 8 12 2)Figure 34 - Tetradecagram *14^4,with signature(1 6 11 2 7 12 3 8 13 4 9 14 5 10 1)Figure 35 - Tetradecagram *14^5,with signature(1 7 13 5 11 3 9 1) (2 8 14 6 12 4 10 2)
  15. 15. The Geometry of Stars© 2013 Tofique FatehiPage 15TABLE OF POLYGRAM SIGNATURESPoints “n” Order “m” Type Signature5 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)
  16. 16. The Geometry of Stars© 2013 Tofique FatehiPage 16SOME OBSERVATIONSThese observations are given without any proofs. In most cases the proofs are self-evident, oreasily derived.In this class of stars, the first thing to notice is that at the core of each polygram is a polygonhaving 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 otherwords, 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 thepolygram can have. This value is also the number of configurations the star can have. We willdesignate “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 aprime number, then the prime factors of “n” will determine which of the “M” configurations areprime, and which are composite. Further, the prime factors of “n” will also determine thecompositions of the composite polygrams.A BRIEF INTRODUCTION TO HYBRID POLYGRAMSConsider the two fourteen-pointed stars *14^3 and *14^5. Both these stars are composite starscomposed 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 thesecond 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 firstorder using all the odd numbered points and interlace it with another septagram of the secondorder inscribed in all the even numbered joints. Its signature is (1 5 9 13 3 7 11 1) (2 8 14 6 12 410 2).
  17. 17. The Geometry of Stars© 2013 Tofique FatehiPage 17Figure 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. Thecentral 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 “hybridpolygrams”.Probably the best way to represent this hybrid tetradecagram is *(2x7)^(1, 2) indicating thatthere 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 theseveral 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 fourprime 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 ordersAnd then to the next prime, and the next … and the next, ad infinitum.
  18. 18. The Geometry of Stars© 2013 Tofique FatehiPage 18But thats not all folks. There are many more classes of stars than the two essayed here. So, asPerry Como sings, “dont let the stars get in your eyes”.Except as otherwise expressly permitted under copyright law, no copying, redistribution, retransmission,publication or commercial exploitation of this material will be permitted without the express permission of thecopyright owner. In the event of any permitted copying, redistribution or publication of copyright material, nochanges in or deletion of author attribution, trademark legend or copyright notice shall be made. Youacknowledge that you do not acquire any ownership rights by downloading this copyrighted material.

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