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A Study of Stars

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Junten Science Library 4 Revised English version based on the lecture on 2nd Dec. 2010 for Junten Junior High School and Chua Chu Kan Higih School in Singapore 星の研究 A Study of Stars Written and translated by Dr. Nakahara －1－ Junten Science Library 4 A Study of Stars §１ An n-pointed Star --- Our Definition of the Star Shape Draw star shapes. What kind of star-shape did you draw? These are the examples of the star shapes I found on the web. Left one is called a pentagram. It is the most common shape almost everyone reminds of when he hears the word “star-shape”. The right one is a hexagram. It is also called the David’s star and is considered as sacred in Judaism. You can find the one on the national flag of Israel (the picture on the right.) I did not expect to find the centre one. It is sometimes adopted by the Manga authors to represent a “girl’s bright eye” or “the star twinkling in the winter sky”. The stars in the real sky are either the planets or fixed stars. The planets wax and wane, changing their shapes. They are by no means of the star shapes. The fixed stars are similar to the sun and are round. It is not the star-shape either. We can not recognize the fixed stars as a disk with definite radius because they are too far from us. They are seen as the bright points. As our photoreceptor cells are of the finite size, the points smear to some shapes which may be star shapes. To investigate the star shapes mathematically, we will have to define them first. Pentagram is a star shape with 5 vertices. Hexagram is the one with 6. When we go further to the decagram, the dodecagram and the icosapentagram, the Latin words for the large numbers are practically undecipherable. Therefore we will use the plainer words with Arabic figures. We call pentagram a 5-pointed star and hexagram a 6-pointed star. These names are less interesting but easier to generalize. Now let us define an n-pointed star. n-pointed star ①n-pointed star is a closed graph(*) with n sides which connects all n vertices of a regular n-gon, the regular polygon(**) with n vertices, and is an Euler’s circuit(***). －2－ ②All the angles are of the same magnitude and all the sides are of the same length. ③No sides are shared with the regular polygon. (*) closed graph A graph with no open segments of line. Every vertex is connected by more than one sides (**)regular polygon A polygon with all the sides are of the same length and all the angles are of the same magnitude. (***)Euler’s circuit A path you can draw by the pencil without lifting it. Exercise 1 Find the one in the above pictures which is a n-pointed star. §2 Drawing n-pointed stars Let us draw various stars with various polygons. Exercise 2 The following are the polygons with 3,4,5,6,7,8,9 and 10 vertices. Draw as many different stars using these polygons. Can you draw stars with any kind of polygons? Can you draw two or more different stars with the same polygon? 3-pointed star 4-pointed star 5 pointed star 6-pointed star 7-pointed star －3－ 8-pointed star 9-pointed star 10-pointed star Let us summarize the exercise. There are no such things as（ ）-pointed stars, （ ）pointed-stars or（ ）-pointed stars. There is only one each of（ ）-pointed star, （ ）-pointed star and （ ）-pointed star. There are two different shapes in （ ）-pointed stars and（ ）-pointed stars. You may have a lot of questions unanswered at the moment. How many polygons with which we cannot draw any stars? We have two polygons with two different kinds of stars. Are they exceptions? Let us proceed to the next polygon and find our idea is not quite right. In fact, there are as many as four different stars we can draw with 11-gon. －4－ Exercise 3 Draw the four different stars with 11-gon. It is now clear that the polygon with many different stars is not the exception at all. For example you can draw 8 different stars with 19-gon. How do we see the number of the different stars with a 360-gon? Our final aim of these lectures is to find the number of the different stars with a given polygon. If the only way we can see is just drawing the stars, our way to the goal will be very long and winding. Is there any systematic method of calculating the number of different stars from the number of vertices? We need to look at the problem from the different angle to make a breakthrough. §3 Stars and Modular Arithmetic ２ １ This is a 7-pointed star. I put the numbers to the vertices of the 7-gon. The vertex on the ３ right is 0. Then the number is getting larger as we move anti-clockwise and the final vertex is 6. Start from vertex 0 and travel along the ０ star sides to the direction shown by an arrow. As the star is an Eulerian circuit, we travel ４ through all the vertices of 7-gon and finally come back to the vertex 0. Let us write down the number in the order of our travel. ６ ５ ０→３→６→２→５→１→４→０ Look at the sequence of the numbers and you will see that the sequence is the same as the one appeared in the multiplication table of modulo 7 arithmetic. －5－ Shown here is the multiplication table of Mod ０ １ ２ ３ ４ ５ ６ the modulo 7 arithmetic. Each entry ７× represents a class of the numbers whose ０ ０ ０ ０ ０ ０ ０ ０ remainders are the same when they are divided by 7. The class 2 contains the １ ０ １ ２ ３ ４ ５ ６ numbers 2, 9, 16, 23 and so on as well as -5, -12, -19. The table is the multiplication ２ ０ ２ ４ ６ １ ３ ５ table. You can find the result of the multiplication among the classes. The ３ ０ ３ ６ ２ ５ １ ４ table shows that the class 4 multiplied by the class 3 makes the class 2, which ４ ０ ４ １ ５ ２ ６ ３ means, a number belonging to the class 4 multiplied by a number belonging to the ５ ０ ５ ３ １ ６ ４ ２ class 3 makes a number belonging to the class 2. The row in pink colour ６ ０ ６ ５ ４ ３ ２ １ represents the multiples of 3, that is:０× ３ １×３ ２×３ ３×３ ４×３ ５×３ ６×３, and is exactly the same as the sequence we have seen in the previous page. The result suggests us that the other rows may correspond to one of the 7-pointed stars. Exercise ４ Look at the other rows of the multiplication table and draw the graphs which correspond to the rows. The result tells us that the row 1 corresponds to 7-gon and the row 2 corresponds to another 7-pointed star which is different from the one corresponded by the row 3. Let us try with the row 4. What kind of shape do we get? The classes attached to the vertices are ０→４→１→５→２→６→３→０ in the order of the path. When we compare this sequence to the one with the row 3, which is ０→３→６→２→５→１→４→０ you will see that the order of the classes is in the opposite direction. Therefore the shapes drawn with these two sequences are exactly the same although the directions we draw them are opposite. －6－ Similarly the star drawn using the row 5 is as same as the one with row 2, and we have the 7-gon again with the row 6 just like with the row 1. Exercise 5 Complete the multiplication table of modulo 10 arithmetic and find the shape drawn using each row. We know that there is only one 10-pointed star. Find the rows which we can draw the 10-pointed star with. Investigate why we can not draw the stars with the other rows. Mod10 ０ １ ２ ３ ４ ５ ６ ７ ８ ９ × ０ １ ２ ３ ４ ５ ６ ７ ８ ９ You saw that in the rows we can not draw the stars the same class appears more than once and certain classes never appear. On the other hand, in the row corresponding to the star every class from 0 up to 9 appears and it does only once. This property means the n-pointed stars are Eulerian circuit. §４ Counting the stars Let us summarize the conclusion of the previous section. (1)Every n-pointed star can be drawn with a row of the multiplication table of modulo n arithmetic connecting the vertices of the n-gon in the order of the number in the row. (2)If we can draw an n-pointed star with a certain row (let us call it the row r) then the same star can be drawn with the row n-r. In this case the pen travels in the opposite direction to the one －7－ with the row r. (3)If a certain row contains all of the classes from 0 up to n-1, then the corresponding shape will be an Eulerian circuit connecting all of the vertices of n-gon.
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