Building Manipulate Demonstrations in Mathematica

Purpose

Manipulate creates interactive visualizations where parameters can be adjusted in real-time. Essential for:

• Building intuition about mathematical relationships
• Creating demonstrations for the dissertation
• Exploring parameter spaces quickly

When to Use

• Visualizing how a function changes with parameters
• Creating educational demonstrations
• Rapid prototyping of ideas before formal analysis
• Exporting interactive HTML for the blog

Basic Syntax

Manipulate[
  expression,
  {variable, min, max}
]

The expression is re-evaluated whenever variable changes.

Example: Simple Case

Manipulate[
  Plot[Sin[n x], {x, 0, 2 Pi}],
  {n, 1, 10, 1}
]

Output: A sine wave with adjustable frequency. The slider goes from 1 to 10 in steps of 1.

Example: Applied to Dissertation

Binary Entropy Function Explorer

Manipulate[
  Module[{h, tangent},
    h[p_] := If[p == 0 || p == 1, 0, -p Log2[p] - (1 - p) Log2[1 - p]];
    tangent = h[p0] + h'[p0] (p - p0) /. p -> x;

    Plot[{h[x], tangent}, {x, 0.001, 0.999},
      PlotRange -> {0, 1.1},
      PlotStyle -> {Blue, {Red, Dashed}},
      AxesLabel -> {"p", "H(p)"},
      PlotLabel -> StringForm["H(``} = `` bits",
        NumberForm[p0, {3, 2}],
        NumberForm[h[p0], {3, 3}]],
      Epilog -> {PointSize[Large], Red, Point[{p0, h[p0]}]}
    ]
  ],
  {{p0, 0.5, "Probability p"}, 0.01, 0.99, Appearance -> "Labeled"},
  ControlPlacement -> Top
]

This creates an interactive exploration of the binary entropy function with:

• Slider for probability $p$
• Tangent line showing the derivative
• Current entropy value displayed

Common Patterns

Pattern 1: Multiple Sliders

Manipulate[
  Plot[a Sin[b x + c], {x, 0, 2 Pi}],
  {{a, 1, "Amplitude"}, 0.1, 2},
  {{b, 1, "Frequency"}, 0.5, 5},
  {{c, 0, "Phase"}, 0, 2 Pi}
]

Pattern 2: Discrete Choices

Manipulate[
  Plot[PDF[dist, x], {x, -5, 10}],
  {{dist, NormalDistribution[], "Distribution"},
   {NormalDistribution[] -> "Normal",
    ExponentialDistribution[1] -> "Exponential",
    PoissonDistribution[3] -> "Poisson"}}
]

Pattern 3: 2D Parameter Space

Manipulate[
  ContourPlot[f[x, y, a, b], {x, -2, 2}, {y, -2, 2}],
  {{a, 1}, -2, 2},
  {{b, 1}, -2, 2},
  ControlType -> Slider2D
]

Pattern 4: Animation-Ready

Manipulate[
  expr,
  {t, 0, 10, AnimationRate -> 1}
]

Click the ”+” to access play/pause controls.

Gotchas

[!warning] Watch Out

1. Slow updates: If expression is computationally expensive, the interface will lag. Use ContinuousAction -> False to only update on release.

2. Variable scoping: Variables inside Manipulate can conflict with global definitions. Wrap complex logic in Module[].

3. Initial values: Always specify sensible defaults: {{x, defaultValue, "Label"}, min, max}

4. Export limitations: Not all features survive HTML export. Test early.

Performance Notes

For expensive computations:

Manipulate[
  expression,
  {param, min, max},
  ContinuousAction -> False,  (* Only update on release *)
  SynchronousUpdating -> False (* Allow interruption *)
]

For very heavy computations, precompute a table and interpolate:

data = Table[ExpensiveFunction[p], {p, 0, 1, 0.01}];
interp = Interpolation[Transpose[{Range[0, 1, 0.01], data}]];

Manipulate[
  Plot[interp[x], {x, 0, 1}],
  {param, 0, 1}
]

Related Tools

• [[Mathematica Plot Customization]]
• [[Exporting Mathematica to HTML]]
• [[Dynamic and DynamicModule]]

References

• Wolfram Documentation: Manipulate
• Wolfram U: “Interactive Visualization” course