Amorous Bugs and Pursuit Problems By Steven Schonefeld

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1 Amorous Bugs and Pursuit Problems By Steven Schonefeld Let us begin with a discussion of the classical Four Bug Problem, where bugs, B1, B2, B3, B4 move in twodimensions so that B1 follows B2, B2 follows B3, B3 follows B4, B4 follows B1. Our bugs are simple-minded each bug moves so it is always pointed at its target bug. We start with the bugs on the corners of a square, as shown below. 1

2 Notation: let [xk, yk] denote the position of bug Bk, k = 1, 2, 3, 4. Of course, x1, y1, x2,.., y4 are functions of t = time. Since B1 is aiming at B2, we have [ x& 1, y& 1] = f ( t)[ x2 x1, y2 y1], where the dot means differentiation with respect to t and f(t) is a scalar function of t. We simplify this problem by assuming that f(t) is identically one, f () t 1. (It turns out that the paths are the same for most any positive f(t).) We have [ x&, y& ] = [ x x, y y ] [ x& 2, y& 2] = [ x3 x2, y3 y2] [ x& 3, y& 3] = [ x4 x3, y4 y3] [ x&, y& ] = [ x x, y y ] We concentrate on the x-coordinates. Equating coordinates gives: &x1 + x1 = x2 (1) &x2 + x2 = x3 (2) &x3 + x3 = x4 (3) &x4 + x4 = x1 (4) 2

3 Differentiating equation (1) gives && x1 + x& 1 = x& 2. Substituting in the left side of equation (2) gives ( ) ( ) x& + x = && x + x& + x& + x = x&& + 2x& + x = x Differentiating this last equation: && x1 + 2x& 1 + x1 = x3, gives &&& x1 + 2&& x1 + x& 1 = x& 3. Substituting in equation (3) gives x& + x = 3 3 (&&& 2&& & ) (&& 2& ) or x + x + x + x + x + x = x &&& x + 3&& x + 3x& + x = x One more differentiation and substitution gives: ( 4) x1 4&&& x1 6&& x1 4x& 1 x1 x1 The auxiliary equation for this last differential equation is = (*) m + 4m + 6m + 4m+ 1= 1 3

4 or 4 ( m + 1) = 1. Solutions to the auxiliary equation are: ( m ) ( m ) ( ) ( ) or + 1 = 1, + 1 = 1, m+ 1 = i, m+ 1 = i m = 0, m = 2, m = 1+ i, m = 1 i. In the complex plane, this set of solutions contains the number zero and the solutions are equally spaced on the circle, radius one, center at 1 + 0i. This will be the case for n = 3 bugs, 5 bugs, 6 bugs, etc. 4

5 Thus the general solution to the differential equation (*) is: x1 = C1 + C2EXP( t) COS( t) + CEXP( t) SIN( t) + CEXP( t) Using Equation (1), we get: x = x& + x Similarly, equations (2) and (3) give x and x 3 4. Similar calculations give the general solutions y1 = D1 + D2EXP( t) COS( t) + D EXP( t) SIN ( t) + D EXP( t) y 3 4 2, y and y We impose the initial conditions: [ x1( 0), y1( 0)] = [ 2, 0] [ x2( 0), y2( 0)] = [ 0, 2] [ x3( 0), y3( 0)] = [ 2, 0] [ x ( 0), y ( 0)] = [ 0, 2] 4 4 5

6 To get C2 = 2, D3 = 2, and all other constants are zero. Thus we have: We may plot the paths of the bugs as below. 6

7 For five bugs, we get similar differential equations. The auxiliary equation for the differential equation in x 1 may be written ( ) 5 m + 1 = 1. This gives five distinct complex solutions, including m = 0, with the rest equally spaced around the circle, radius one, center at 1 + 0i. We get the paths of five bugs, shown below. 7

8 The path of each bug is said to be a logarithmic spiral. Paths of three bugs. 8

9 Paths of six bugs. 9

10 Paths of seven bugs. With the above patterns, n bugs start on the vertices of a regular n-gon with center at the origin so that each vertex is distance two from the origin. For convenience, bugs are numbered in a counter-clockwise manner and we always have bug B1 start at the point (2, 0). We will call the above patterns the standard patterns. With the standard pattern, the distance traveled by each of the n bugs is the same. With the help of DERIVE, we may find the arc length of the path of B1 for a large number of values of n. This leads us to conjecture that the distance 10

11 traveled by each of the n bugs in the standard pattern is given by: L = 2 n sin π n = 3, 4, 5, 6, 7,... n 11

12 SKIP-ONE PATTERNS. These patterns are sometimes said to have a star-shaped initial polygon. As with the standard pattern, the bugs start on a regular n-gon with B1 at the point (2, 0), but B2 is the second bug away from B1, B3 is the second bug away from B2, etc. See the figure below for n = 5. 12

13 Skip-one for n = 7. 13

14 Skip-one for n = 9. As above, we may conjecture the distance traveled by each bug in the skip-one pattern to be: L n = 2 2 sin n π, for n = 5, 7, 9,.. 2k+1 Note: the conditions for possible n in the skip-one pattern are: (1) n 5, (2) n cannot be a multiple of two. 14

15 SKIP TWO PATTERNS Here again, the bugs start on a regular n-gon, but B2 is the third bug away from B1, B3 is the third bug away from B2, etc. See the figure below for n = 7. 15

16 Skip-two pattern for n = 8. Here we include the starshaped initial polynomial. 16

17 Skip-two pattern for n =

18 Skip-two pattern for n =

19 Skip-two pattern for n = 13. We may conjecture the distance traveled by each bug in the skip-two pattern to be: L n = 2 3 sin π for n = 7, 8, 10, 11, 13, 14,.. n Note: the conditions for possible n in the skip-two pattern are: (3) n 7, (4) n cannot be a multiple of three. 19

20 Bugs Flying in Three Dimensions Our work is more complicated in three dimensions. Here is a picture depicting Four Bugs (starting at the vertices of a regular tetrahedron) in three dimensions moving (as usual) so that B1 follows B2, B2 follows B3, B3 follows B4, B4 follows B1. 20

21 It may be easier to see the paths if they are enclosed in space tubes (which shrink in diameter as t increases) as shown below. 21

22 Six Flying Bugs Initial points 22

23 Eight Flying Bugs starting on the corners of a cube. 23

24 Eight Flying Bugs with Different Starting Conditions. 24

25 Initial Conditions for this last plot References for Bug Pursuit Problems Perhaps the best way to find references on this problem is to do an Internet search for: Bugs Pursuit Problems Following are a few references. Gardner, M. The Scientific American Book of Mathematical Puzzles and Diversions. New York: NY: Simon and Schuster, Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp ,

26 Good, I. J. "Pursuit Curves and Mathematical Art." Math. Gaz. 43, 34-35, Lucas, E. "Problem of the Three Dogs." Nouv. Corresp. Math. 3, , (This may be the first mention of this type of pursuit problem!) Peterson, I Pursuing pursuit curves. Science News Online (July 14). Available at also E. W. Weisstein, Mice Problem, MathWorld--A Wolfram Web Resource 26

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