If you have any difficulty running them, try this.
This program is for the '1089 trick' itself, introduced on pages 1 and 2 of the book.
This program animates the figure on page 45, and shows, in particular, the variation in speed as the planet or comet proceeds around its orbit.
Various
ways of calculating pi
This program compares the four methods of calculating pi that are
mentioned on pages 88-90, namely Viete's product formula, Wallis's product
formula, Leibniz's series and Euler's series.
For an alternative program, for large numbers of terms or
factors, click
here
.
This program shows the first few modes of vibration of a stretched string, as on pages 100-101.
Strange behaviour of an infinite series
This program is concerned with how the infinite series
1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + .............
can be made to converge to different
'sums' by adding the terms in different orders.
The user can choose the order freely, and see the results graphically.
For an alternative program, more suitable for large numbers of terms,
click
here.
Computer solution of differential equations
This software shows computer 'solutions' for the bobbing spider problem on page 116. The smaller the time step, the more accurate the solution.
This compares the two different methods for calculating e which are discussed on pages 124 and 133.
Using
playing cards to determine e
This program uses a random number generator to simulate the playing-card
routine described on page 131.
For an alternative program, more suitable for large numbers of trials,
click here.
The
gravitational three-body problem
This program includes an animation of the figures on pages 136-7, showing
the (complicated) motion of three equal masses which attract one another
according to the law of gravitation.
While the three-body problem is a very old one, some very interesting
'simple' motions have been discovered only recently, and the software
includes an example of one of these. (But see also the external software
listed further down this page.)
An
elementary example of chaos
This program relates to the 'simple' chaotic system discussed on pages 140-141. The user can experiment with different values of the parameter a, and different starting values x1. The outcome is typically chaotic if a > 3.57 .
Chaos
and catastrophe for a vibrated pendulum
This program is an animated version of the figure on page 145. The pivot of a simple, rigid pendulum is vibrated up and down, in an entirely regular way, at twice the pendulum's natural swinging frequency. The user can gradually alter the magnitude A of the pivot motion. Striking changes in the behaviour of the pendulum take place as A is gradually increased - and, indeed, when A is gradually decreased again.
The
upside-down pendulum theorem
This program illustrates Chapter 15: 'Not Quite the Indian Rope Trick', with a gravity-defying system involving three linked pendulums, turned upside-down.
The proof of Pythagoras's theorem on page 11 is very neat and simple, but Euclid's original proof is different, and based instead directly on the figure on page 10. The link above offers an attractive, 'animated', version of Euclid's proof.
This Java applet offers an animated version of the figure illustrating elliptical planetary motion on page 45. By simply adjusting the slider at the bottom, the degree of ellipticity can be changed. A highly elliptical orbit (as for a comet) means very substantial increases in speed as the object gets closer to the 'Sun'.
This is a computer simulation of the problem discussed on pages 67-69. The user selects an end-point for the wire, and the program simulates the motion for four different shapes of wire. The one shaped as a cycloid (green) always gives the shortest time.
Dropping
needles to determine pi
This is a computer simulation, using a random number generator, of the 'dropping needles' method for estimating pi, described on page 92. If you press the 'Drop 1000' button several times in succession it's quite fast, but I'm not sure it's as much fun as drawing some lines on a sheet of paper and doing it for real!
New developments in the
n-body problem
Although motions in the gravitational N-body problem are typically chaotic
(see the case N=3 on pages 136-7), some remarkable periodic motions have
been discovered very
recently, and this link provides some impressive animations.
A nice
article on all this appeared in the New Scientist magazine in
August 2001.