math.gif (2730
                          bytes) Mathematics


A. Reference D. Functions G. Wavelets
B. General & Miscellaneous E. Fourier Analysis, FFTs H. Cryptography
C. Numerical Methods F. Fractals and Chaos  

also see efg's Delphi Math Info and Links, Delphi Math Functions, Math Projects

A. Reference

Calculators The Calculator Home Page. has a number of calculator related resources, including an online scientific calculator, units conversion and constants database, and other information. 

Geometry Algorithms 



Famous Curves 

2D Curves 

Geometry, Solid geomlogo.gif (987 bytes)The 5 Platonic Solids and the 13 Archimedean Solids
Mathematics Library/Archive

Front for the XXX Mathematics Archive
UC Davis seach engine of xxx mathematics archives at Los Alamos

Math Stuff Library by Don Lancaster.  Included are such topics as Fourier Series,  linear  algebraic equations, nonlinear graphic transforms, chain codes, digital filters, halftone techniques, and similar subjects.

Mathematics Archives WWW Server

Mathematics links, 

Sloane's On-Line Encyclopedia of Integer Sequences Look up a number sequence in the database
WebMath Students, get help with your homework
World of Mathematics Eric Weisstein's World of Mathematics
The Web's Most Extensive Mathematics Resource
(offline now because of a legal fight between CRC Press and Eric Weisstein)

B. General & Miscellaneous

American Mathematical Society
Boids Coordinated animal motion such as bird flocks and fish schools 
Decision Tree for Optimization Software  
Forensic Mathematics Mostly about DNA identification 
Four-Color Theorem

Four-Color Theorem   

The Four Color Map Problem 

Fun with Mathematics Harry J. Smith's page:  Pi, Tesseract, Gaussian Primes, Fibonacci, Ackermann's Function, Encyrption, Perfect Numbers, Mersenne Primes, Quaternions,  much more
Graphics for the Calculus Classroom

Computational Geometry Algorithms Library.  The CGAL project is a collaborative effort to develop a robust, easy to use, and efficient C++ software library of geometric data structures and algorithms. 

Geometric Paper Folding

Geometry in Action

Paper and Plotter: A 3D Surface

Polyhedra Database

Geometery Junkyard

Klein Bottle Shown here is a glass "Klein Bottle" made in the 70s by the late Mitsugi Ohno (Mr. Ohno died on Oct. 22, 1999 from cancer), an expert glass blower at Kansas State University. A Klein Bottle is a 3D Möbius Strip. See the article "Glass Klein Bottles" in the March 1998 Scientific American "Mathematical Recreations" section (pp. 100-101) for more information. 

Mitsugi Ohno is truly a master artist.

Math Software
Mathematical Experiences through Image Processing
Mathematical Markup Language A low-level specification for describing mathematics as a basis for machine to machine communication. It provides a much needed foundation for the inclusion of mathematical expressions in Web pages.  Specification:
Mathematics & Science
Mathematics Renderers Comparative Review of World-Wide-Web Mathematics Renderers
Maths Is Fun 

                    (1033 bytes)   efg's Calculate Pi Using Buffon's Needles Lab Report

On the computation of the n'th decimal digit of various transcendental numbers

Prime Numbers The Prime Pages

The First 10,000 Primes

Great Internet Mersenne Prime Search (26,972,593 - 1 is prime)

On August 31, 1998, Ray Ballinger has found a new largest known
twin primes record with "Proth" written by Yves Gallot:
835335*239014-1 and 835335*239014+1 (11751 digits).
For more details, see

David Kuestler's UseNet Post with Prime Numbers Generator C code "erato"
(sieve of Eratosthenes for up to 2^32 by Jim Gillogly)

Ram's Horn Cardboard model of a logarithmic spiral or maybe helix would be a better word

Ram's Math

Random Numbers Server on the Theory and Practice of Random Number Generators

Papers by Pierre L'Ecuyer about random number generation

Rounding Rules On the Standard Rounding Rule for Multiplication and Division
Significant Figures Significant Figures and Rounding Rules
Sound of Mathematics
Slide Rule Universe
Statistics & Probability Engineering Statistics 

VassarStats:  Web Site for Statistical Computation

Stony Brook Algorithm Repository
Comprehensive collection of algorithm implementations of
fundamental problems in combinatorial algorithms.

Symbolic Computation JACAL is an interactive symbolic mathematics program. JACAL can manipulate and simplify equations, scalars, vectors, and matrices of single and multiple valued algebraic expressions containing numbers, variables, radicals, and algebraic differential, and holonomic functions.

C. Numerical Methods

Also see
NIST Guide to Available Mathematical Software

ACM Collected Algorithms from Transactions on Mathematical Software Nearly 40 years of ACM publications on mathematical algorithms.

.GZ file Format:

ASCII Format:
Approximation Series Approximation Techniques 
Complex Roots 
Electronic Library (eLib) for Mathematical Software
GAMS: Guide to Available Mathematical Software U.S. National Institute of Standards and Technology (NIST).
A cross-index and virtual repository of mathematical and statistical software components for use in computational science and engineering.
MAGIC "My Alternate Graphics and Image Code" by Dave Eberly. Includes several programs about numerical methods including eigensystems, integration, ordinary differential equations, root finders, minimizers,  linear system solvers, and special functions.
Netlib Repository Collection of mathematical software, papers, and databases.

Netlib Master Index
Numerical Recipes (Book) Home Page:


Roots of a Real Function 

Template Numerical Toolkit The Template Numerical Toolkit is a collection of mathematical libraries for numeric computation in C++. Its fundamental classes include vectors, matrices, and multidimensional arrays.

D. Functions
Also see efg's Delphi Math Functions Pages

Wolfram's Functions Site

Approximations Functional Approximations 
Bessel Functions Bessel Programs 

Bessel Function

Symbolic and Numerical Computation on Bessel Functions of Complex Argument and Large Magnitude,


The Scientist and Engineer's Guide to Digital Signal Processing
Book Home Page: including PDF files

gamma (with graph)


Books about Mathematical Functions

An Atlas of Functions by Spanier and Oldham. Hemisphere Publishing, 1987.

CRC Handbook of Mathematical Curves and Surfaces, CRC Press, 1990.  (out of print)

CRC Standard Mathematical Tables, CRC Press, any version.

Handbook of Mathematical Functions by Abramowitz and Stegun. Dover Publications, 1972.

Numerical Recipes in C: The Art of Scientific Computing (Second Edition), William H.Press, et al, Cambridge University Press, 1992. (Also available in Pascal, FORTRAN, and Basic programming languages.)

Special Functions & Their Applications, Lebedev. Dover Publications, 1972.

Tables of Functions, Jahnke and Emde. Dover Publications, 1945.

E. Fourier Analysis, FFTs

Books Chapter 5, Processing Images in Frequency Space
The Image Processing Handbook
Complex Fast Fourier Transform The Scientist and Engineer's Guide to Digital Signal Processing
Cross Correlation sci.image.processing post by Chris Russ (29 Sept 99):  "Cross correlation in the Fourier domain is really the same as convolution with a 180 degree rotated pattern in the spatial domain.  If you're looking for speed and that is the reason you're interested in an FFT-based implementation of cross-correlation, you may want to consider this equivalence."

"A small spatial convolution (say under 13x13) is less numerically intensive (and usually faster) than two forward FT's a complex multiplication with the phase reversed and an inverse FT."

Fast Fourier Transform Code and Information
Fast Fourier Transform Algorithm
FFT code and related stuff
FFT Demystified 
FFTW FFTW is a C subroutine library for computing the Discrete Fourier Transform (DFT) in one or more dimensions, of both real and complex data, and of arbitrary input size. We believe that FFTW, which is free software, should become the FFT library of choice for most applications.  Developed at MIT.   [Winner of 1999 Wilkinson Prize for Numerical Software -- awarded by Argonne Nat'l Lab (U.S.),   National Physical Laboratory (UK), Numerical Algorithms Group (UK)]
Fourier Theory
Fourier Transform in 2D
Fourier Transform in 2 dimensions
HpW Works The HpW Works is spectrum Analyzer for MS Windows on the Intel platform. The Analyzer uses the Fast Fourier Transformation (FFT) technique to transform the sampled data to the frequency domain. It is a way of representing a very complicated sampled data pattern in terms of its linear projection onto sinusoids of various frequency. The FFT spectrum is a power frequency graph which shows quantitatively how much of the original pattern is composed by each of the frequencies.
Image Transforms Chapter 3, pp. 81-159
Digital Image Processing
, 2nd edition
Signal Proecessing FFT, Butterworth Filter, ... 
TFastFourier Delphi Component,

F. Fractals and Chaos
Also see efg's Fractals and Chaos Projects Page

general Dr. G. A. Edgar
3D Fractal Landscapes
Animated GIFs
Chaos Theory (Auf Deutsch)
Clifford Pickover's Web Page. Check out descriptions of Cliff's many books.
Dynamical Systems The Dynamical Systems and Technology Project at Boston University
FAQ Fractal Frequently Asked Questions and Answers 
Generating Random Fractal Terrain
IFS Iterated Function Systems

IFS Fractals Digital Encyclopedy   (English or French)

Nonlinear Iterated Function Systems  

Keplerian Fractals Koch Snowflake -- very interesting variations
Links Fractal Music
Several dozen links in UseNet Post by Paul N. Lee
Julia and Mandelbrot Sets On Gaston Julia's birthday on 3 Feb 2004, Google showed this logo on their search page:

Gaston Maurice Julia

The Mandelbrot and Julia sets Anatomy Contents

  efg's Fractals Show 2 Lab Report
Create Mandelbrot and Julia sets for a number of complex math functions, including z2, zn, zx, zz, exp(z), cos(z), sin(z), cosh(z), sinh(z), I0(z), J0(z), tan(z), tanh(z), sec(z), sech(z), csc(z), csch(z), cot(z), coth(z), gamma(z).

Les Paysages Fractals IFS
Lyapunov Exponents lyapunov.gif (16030 bytes) efg's Lyapunov Exponents Lab Report
This project describes how to compute and display the Lyapunov exponents of the logistic map with periodic forcing (and to compute and display some very interesting graphics images).
Meus Fontes em Pascal LSM, TCAOS, Logic Chaos, Fractal.EXE
Mandelbrot Benoit Mandelbrot's Home Page 
Mandelbrot Sets See Julia and Mandelbrot Sets (above)

"The Area of the Mandelbrot Set" by John H. Ewing and Glenn Schober
Numerische Mathematik, 61, 59-72 (1992).

Pi and the Mandelbrot Set  

Multifractal Walk Down Wall Street by Benoit B. Mandelbrot
Scientific American, February 1999, pp. 70-73
Sierpinski Gasket Triangle Sierpinski.gif (2621 bytes)   efg's Sierpinski Triangle Lab Report
This project shows how to create a Sierpinksi gasket, a "holey" triangle, by recursively cutting holes in a triangle.
Sprott's Fractal Gallery
von Koch Curve VonKochSnowflake.gif (1087 bytes) efg's von Koch Curve Lab Report
This project shows how to create a von Koch curve, including a von Koch snowflake.

G. Wavelets

Algorithms Wavelets offer an attractive alternative when it comes to image compression. Steven gives us a practical guide to image compression via wavelets, and compares the results to more familiar techniques, such as JPEG. Dr. Dobb's Journal, August 1999.
Amara Graps

Amara's Wavelet Page

Books Wavelet Analysis with Application to Image Processing

Wavelets for Computer Graphics

Chapter 7, Wavelets
Algorithms for Image Processing and Computer Vision

MR/1 Software Package for Multiresolution Analysis 

Book:  Image and Data Analysis: The Multiscale Approach, Cambridge University Press, 1998.

Wavelet Digest and E-Mail newsletter
Wavelet NetCare
Wavelets Library By Don Lancaster

H. Cryptography
Also see efg's Delphi Cryptography and Multiple-Precision Arithmetic Page

general sci.crypt FAQ 

Cypherpunks, Cryptography & Hackers

Crypto Drop Box
Government Encryption Standard DES Takes a Fall
Congratulations, Rocke!
Great Internet Mersenne Prime Search
Microsoft's Cryptography Research page
Quadralay's Cryptography Archive
Two Fish The Twofish Encryption Algorithm, Dr. Dobb's Journal, December 1998.
Wassenaar Arrangement Agreement of 33 member countries to restrict exports of 56-bit general encryption products,

Updated 17 Dec 2017
Since 1 Nov 1998