Visualizing hexagons: An interactive approach to hue and shape in Wolfram Mathematica

[Splendid Digital Solutions]

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byu/DigitalSplendid inMathematica

Disclaimer: This article was created with the assistance of an AI language model and is intended for informational purposes only. Please verify any technical details before implementation.

Certainly! Let’s break down the code and the logic behind each part to understand how this `Manipulate` creates a variable number of hexagons with a consistent, adjustable hue.

Here’s the full code:

“`wolfram
Manipulate[
Graphics[
Table[
{Hue[Mod[hue, 1]],
Polygon[Table[{Cos[θ] + i*1.5, Sin[θ]}, {θ, 0, 2 Pi, 2 Pi/6}]]},
{i, 0, numHexagons – 1}
]
],
{{numHexagons, 5, “Number of Hexagons”}, 1, 10, 1},
{{hue, 0, “Hue”}, 0, 1, 0.01}
]
“`

Let’s go through each section step by step.

### 1. `Manipulate`

The `Manipulate` function in Wolfram Mathematica is used to create interactive controls for dynamic visualizations. Here, we’re using it to adjust:

– The **number of hexagons** displayed (`numHexagons`).
– The **color (hue)** of each hexagon.

Syntax:
“`wolfram
Manipulate[ content, {control1, min1, max1, step1}, {control2, min2, max2, step2} ]
“`

– `content`: The main expression (a `Graphics` object in this case) that will dynamically change based on the values of the controls.
– `{control, min, max, step}`: Definitions for interactive sliders or input fields for each control.

### 2. `Graphics`

The `Graphics` function is used to generate and display graphical shapes, in this case, hexagons. This part holds the geometric shapes we want to visualize.

### 3. `Table`

The `Table` function is used to generate a list of items. Here, it’s used twice:

1. To create multiple hexagons based on the value of `numHexagons`.
2. To define each hexagon’s vertices using trigonometric functions for its regular shape.

Syntax:
“`wolfram
Table[expression, {variable, start, end, step}]
“`

For our hexagon setup:
– We create a set of hexagons with `{i, 0, numHexagons – 1}`, where `i` is the index for each hexagon. This will run `numHexagons` times, producing one hexagon per iteration.

### 4. `Hue[Mod[hue, 1]]`

This line controls the color of each hexagon by applying the `Hue` function, which generates colors based on the hue value (ranging from 0 to 1).

– **`hue` Slider**: We use the slider value (`hue`) to control the color. Changing `hue` adjusts the color of all hexagons simultaneously.
– **`Mod[hue, 1]`**: Ensures that even if `hue` goes slightly over 1 due to rounding or small adjustments, it wraps back to a valid hue value within `[0, 1]`. The modulo operation keeps the hue value within the color spectrum bounds.

### 5. `Polygon[Table[{Cos[θ] + i*1.5, Sin[θ]}, {θ, 0, 2 Pi, 2 Pi/6}]]`

This part constructs each hexagon:

– **`Table[{Cos[θ], Sin[θ]}, {θ, 0, 2 Pi, 2 Pi/6}]`**:
– This inner `Table` generates six equally spaced points around a circle by calculating the coordinates `{Cos[θ], Sin[θ]}` for angles between `0` and `2π` at intervals of `2π/6` (360°/6 = 60°, one point per corner of the hexagon).

– **`+ i*1.5` Offset**:
– `{Cos[θ] + i*1.5, Sin[θ]}` shifts each hexagon to the right by `i * 1.5`. This avoids overlap, as each hexagon’s center is moved `1.5` units to the right of the previous one. By adjusting this offset, we position each hexagon side-by-side.

### 6. Controls within `Manipulate`

The final part sets up the interactive sliders:

– **`{numHexagons, 5, “Number of Hexagons”}, 1, 10, 1`**:
– This slider lets users choose the number of hexagons to display, from `1` to `10`. The slider starts at `5` hexagons by default, with a step size of `1`.

– **`{hue, 0, “Hue”}, 0, 1, 0.01`**:
– This slider allows the user to control the color of all hexagons. It ranges from `0` to `1` with a step size of `0.01`, giving a smooth color adjustment over the hue spectrum.

### Summary

– **Changing `numHexagons`**: Adjusting this value increases or decreases the number of hexagons. Each hexagon is positioned side-by-side due to the `i*1.5` offset.
– **Adjusting `hue`**: Modifying the hue slider changes the color of all hexagons simultaneously.

This structure provides control over the number and color of hexagons while ensuring that each hexagon is consistently spaced and shares the same color as determined by the `hue` value.

 

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