IntroductionMathgrapher version 2 has just been released. You are invited to download a fully functional trial version. Install Mathgrapher, start the Demonstrations and see what Mathgrapher can do for you. Functions in 2D and 3D MathGrapher is a graphical calculator for functions of the form F(x) and F(x,y) containing up to 20 subfunctions and 150 numerical and 100 named constants. Cartesian as well as polar coordinates can be chosen and functions can be represented in patametrized form (2D). F(x,y) can be represented in 2D and 3D by Shaded surfaces, Contour plots and Cross-sections through Contourplots. In the 3D viewer you may rapidly vary the viewing angle, distance and shading of the 3D surface using your mouse. Data in 2D and 3D Edit and draw graphs of your 2D or 3D Data. 3D surfaces can be previewed in the 3D viewer (OpenGL). Shaded surfaces, Contour plots and Cross-sections through Contour plots can be drawn in same way as 3D Functions. Curve fitting (linear and non-linear) A number of least squares curve fitting methods can be selected: e.g. linear regression, polynomials, trigonometric polynomials and cubic splines. An important feature of this program is that you can use the general and powerfull (non-linear) Levenberg-Marquardt method to fit your data to any continuous function you define. Iterations Calculate algebraic series or study iterative multi-dimensional maps. Several mathematical tools are available to analyse the results (zie ODE's below). Look at the examples to see how you may use Mathgrapher to study the route to chaos via period doublings in the simple logistic mapOrdinary Differential Equations (ODE's) The evolution of dynamical systems in physics, chemistry, electronics, economics and population dynamics can often be described with a set of coupled ODE's. Mathgrapher uses an accurate Adams-Bashforth variable order, variable step predictor-corrector algorithm to integrate systems of up to 20 coupled ODE's. Several tools are available to analyse the results of the integrations (and iterations) such as: Graph of the time evolution, Projections in 2 or 3 dimensions, Surfaces of Section and Power spectrum analysis.

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Lindenmayer systems

Lindenmayer systems (L-systems, turtle graphics) are used to generate Fractal curves (Koch, Sierpinski, Levy, Dragon), Space filling curves (Hilbert, Peano-Gosper), Growth patterns in plants, etc. It is an iterative string rewriting method. Starting from the Axioma a new string is formed by replacing the string symbols according to a few given Rules.The final string contains instructions telling the turtle where to go, or how to draw the path.
The list contains a number of well known examples. Choose one of them and push the Draw button to draw it in the graphical window. You may construct your own examples and add them to the list.
More information on Lindenmayer systems:

Some history Mathforum Fractint Ridldle Wikipedia Mathworld Algorithmic botany

L-systems: Examples - Fractals

Lindenmayer systems - Fractal curves

Koch curve for order 1 to 4. The line length is adjusted (27, 9, 3,1; see the Demonstration)

Koch's snowflake:

Sierpinski Gasket:

Sierpinski Carpet:

Minkowski:

The Dragon:

Archipel (Mandelbrot):