π π π points:= { seq([ncos(pi/4n), Nsin(Pi/4N)], N=0..120) }:

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Kory Sibyl Spencer
6 years ago
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1 ITEGER SPIRAL Several years ago we found a way to conveniently plot all positive integers as points along an Archimedes spiral. We want here to examine this problem in more detail. We begin by defining a complex function- π π π f( ) = exp( i ) = cos( ) + isin( ) One can generate a sequence of complex numbers f() by evaluating things over the range 0<<0. These numbers are easy to generate by the MAPLE operation- points:= { seq([*cos(pi/4*), *sin(pi/4*)], =0..0) }: ext these 0 points are plotted in the complex plane x-y we also superimpose the Archimedes Spiral r=(4/π)θ. The comms for doing this are- A:=pointplot(points,symbol=circle,color=red,axes=none): B:=plot(4*/Pi,=0..30*Pi,coords=polar,color=blue): followed by- display(a,b,scaling=constrained,title=`iteger SPIRAL`);

2 Here is the result- The red circles represent the positive integers while the blue spiral represents the Archimedes Spiral. ote that the integers all are located at the intersection of the spiral one of eight straight lines shown in black. All even numbers lie along the x y axis while all odd numbers are located along the diagonals x=y or x=-y. There are eight categories of numbers defined by the value of k in the number = 8n+k, k=0.. n an integer. The value of k is easily determined by dividing the integer by 8 then looking at the remainder. Thus we can rewrite the number- 459 = 8(303) + In modular arithmetic language this is the same as saying- 459 mod8=

3 It indicates that 459 lies along the diagonal in the 4 th quadrant at the 303 rd turn of the spiral. The mod operation is built into most canned mathematics programs such as MAPLE or MATHEMATICA. Two numbers lying along the same straight line will always be separated from each other by multiples of 8. Also note that (except for =) all prime numbers lie along one of the four diagonals. This fact has implications for the meaning of the Ulam prime spiral which(as we have shown earlier) is really just a morphed version of the present integer spiral its pattern really says nothing more than that all prime numbers are odd. Two important set of numbers which produce integer values are the Fermat umbers the Mersenne umbers. They are defined as - F( n) n + M( n) n+ for n=,,3, 4... These numbers will be prime for certain values of n. For example, F(3)=5 M()= are prime. However, F(5)= M()=04 are composite. To find were the numbers F(n) M(n) are located within the x-y plane one simply performs the mod operation. For F(n) this produces a remainder of one means that all Fermat numbers lie along the diagonal in the first quadrant. For example,- F(4) 6 + = 6553= 8(99) + F (6 ) 64 + = 8( ) + ote also that, as expected, the difference F(6)-F(4) is a multiple of 8. All Mersenne umbers above n= show a remainder of after division by 8. Thus Mersenne numbers, including those which are prime, all lie along the diagonal in the 4 th quadrant differ from each other by multiples of eight. Thus- 3 M (6) = 8(03) +, M() = 8(4095) + M (8) = 8(6383) +, M(9) 9 = 8(65535) +

4 We also observe that all these Mersenne numbers end in either or when expressing things in a decimal number system. In binary these numbers are represented by a series of all s no 0s, thus all end in. One may use modular arithmetic to carry out basic operations involving integers. Take the case of addition using the two specific numbers- 3 = + = = {8(48) + 3} + {8(86) + 5} = 8(89) + 8= 030 so 030mod8= 0 It shows that the sum 3 lies along the positive x axis at the 90 th turn of the spiral. If we perform a product operation of these numbers, we find- 4 = = {8(48) + 3}{8(86) + 5} = 64(48)986) + [40(48) + 4(86)] + (3)(5) mod(8) 363= 8(9588) + 363mod8= The product thus lies along the diagonal in the 4 th quadrant. We can also ask where will the large number = lie. The answer is along the diagonal in the 3rd quadrant since mod8= The digit prime number = yields mod8= so lies along the diagonal in the st quadrant. With the mod 8 operation one can also quickly tell in which quadrant a number lies. Take,for example, the product = 4 6 It lies along the negative y axis at y= Finally we look at what happens when is non-integer. Keeping the same definition used for integer, we can express, for example, the square root of two as- 5

5 π π {cos( ) + sin( )} = i So that root two lies at a radial distance sqrt() from the origin at angle θ=arctan(.09886)=63.65 deg in the first quadrant. The neighboring number =3/ will lie at radial distance.5 angle 3π/8rad=6.5deg=(90+45)/deg. Using the mod operation we can also pinpoint the location of a non-integer such as =34.95 as follows- = 34.95= 348 mod with 348 mod8= 4 Thus lies in the second quadrant at radial distance from the origin at angle θ=9π/80rad=.5deg. January 0

RELATIONSHIP BETWEEN INTEGER SPIRALS AND SPIDER WEBS

RELATIONSHIP BETWEEN INTEGER SPIRALS AND SPIDER WEBS In the last several years we have-examined certain unwinding spirals which are intersected by radial lines passing through the spiral origin. We have