mathematica

Let’s play the following “chaos game”. Consider a regular triangle and take some initial point in it. Now pick one of the three vertices at random, connect it with the current point, and mark this line’s midpoint as the new point. If you repeat this process many times, what will the resulting shape look like? You might expect to find the points simply shattered everywhere, without much structure, but surprisingly, the points will in fact create a Sierpinski fractal.

There are lots of variations on this theme, some of which can be found on this Wikipedia page.

Animation made in Mathematica.

n roots of unity

When one is dealing with complex numbers, it is many a times useful to think of them as transformations. The problem at hand is to find the n roots of unity. i.e

As is common knowledge z = 1 is always a solution.


Multiplication as a transformation

Multiplication in the complex plane is mere rotation and scaling. i.e

Now what does finding the n roots of unity mean?

If you start at 1 and perform n equal rotations( because multiplication is nothing but rotation + scaling ), you should again end up at 1.

We just need to find the complex numbers that do this.i.e

This implies that :

And therefore :

Take a circle, slice it into n equal parts and voila you have your n roots of unity.


Okay, but what does this imply ?

Multiplication by 1 is a 360o / 0o rotation.

When you say that you are multiplying a positive real number(say 1) with 1 , we get  a number(1) that is on the same positive real axis.


Multiplication by (-1) is a 180o rotation.

When you multiply a positive real number (say 1) with -1, then we get a number (-1) that is on the negative real axis 

The act of multiplying 1 by (-1) has resulted in a 180o transformation. And doing it again gets us back to 1.



Multiplication by i is a 90o rotation.

Similarly multiplying by i takes 1 from real axis to the imaginary axis, which is a 90o rotation. 

This applies to -i as well.

so on and so forth,

Have a great day!

6

A Fascinating Calculation

Dividing by 7 yields this fascinating play of the numbers 1,4,2,8,5 and7. I am not going to spoil the fun by letting you on the pattern that emerges and other captivating properties that you might discover along the way.

Although feel free to write to us if you found anything that marveled you and we would definitely share it with the world.

Who knew division could be this much fun. am i right?

Have fun!

4

Algorithmically generated using circles subjected to constrained randomness

Animated GIFs (600x600)

Mathematica code:

RR[n_, m_] := (SeedRandom[n*m]; RandomReal[])

L[t_, Q_, g_, i_] := Sum[ Exp[-(t - (RR[i, 5*Q] + j))^2/g], {j, {-1, 0, 1}}]

G[c_, F_, P_, s_, o_, g_, A_, a_, w_, t_, r_, Q_, pr_, is_] :=
Graphics[
 Table[
  Table[
   {RGBColor[RR[i, Q], RR[i, 2*Q], RR[i, 3*Q], o + A*L[t, 5*Q, g, i]],
    Disk[
     RotationTransform[(k + (-1)^(i*w)*t*a)*2 Pi/F]
      [{r*RR[2 i, 4*Q], r*RR[2 i + 1, 4*Q]}], s*L[t, 5*Q, g, i]]},
  {k, 0, F - 1, 1}],
 {i, 1, P, 1}],
PlotRange -> pr, ImageSize -> is,  Background -> c]

 
Manipulate[[
  G[RGBColor[54/255, 70/255, 93/255], 12, 50, .2, 0, .1, .25, 1, 1, t, 1, Q, 1.6, 600],
 {Q, {41, 72, 1, 76}}, {t, 0, .96, .04}],
List of Free Science Books

Here’s an alphabetical list of all available free books. Note that many of the links will bring you to an external page, usually with more info about the book and the download links. Also, the links are updated as frequently as possible, however some of them might be broken. Broken links are constantly being fixed. In case you want to report a broken link, or a link that violates copyrights, use the contact form

A

  • A Beginner’s Guide to Mathematica
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  • A First Course in General Relativity
  • A New Astronomy
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  • A Popular History of Astronomy During the Nineteenth Century, Fourth Edition
  • A Review of General Chemistry
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  • AMPL:  A Modeling Language for Mathematical Programming
  • An Introduction to Elementary Particles
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  • An Introduction to Mathematics
  • An Introduction to Proofs and the Mathematical Vernacular
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B

  • Basic Algebra, Topology and Differential Calculus
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C

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D

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E

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F

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G

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H

  • Handbook of Formulae and Physical Constants
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I

  • Intelligent Systems
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K

  • Kinetic Theory

L

  • Laboratory Manual for Introductory Physics
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M

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N

  • Natural Disasters
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O

  • Observing the Sky from 30S
  • On Particle Physics
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P

  • Particle Physics Course Univ. Cape Town
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Q

  • Quantum Dissipative Systems
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R

  • Recreations in Astronomy
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S

  • Short History of Astronomy
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T

  • The Astrobiology Primer: An Outline of General Knowledge
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  • The Zij as-Sanjari of Gregory Chioniades (June 27, 2009)
  • Three Dimensional Geometry

U

  • Understanding Physics
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