Exploring the Concept of Dimensional Reduction

Higher Dimensions & Projection

In physics, we often describe space using three spatial dimensions (x, y, z) and one time dimension (t). However, mathematics allows for spaces with more than these four dimensions. These higher-dimensional spaces are difficult to visualize directly.

When projecting a 3D object onto a 2D plane (like a screen), information is inevitably lost. The projection creates a distortion – a flattening – of the original shape.

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The Geometry of Collapse

Dimensional collapse simulates this process by representing a higher-dimensional object (e.g., a 4D hypercube, or tesseract) as a lower-dimensional projection – typically 3D or 2D.

This is achieved through a mathematical transformation that maps points in the higher dimension to corresponding points in the lower dimension while attempting to preserve some key geometric properties.

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Visualizing the Reduction

The simulation allows you to manipulate the projection parameters, such as the angle of view and the dimensionality reduction factor. You’ll observe how different choices affect the appearance of the collapsing object.

Observe that the perceived shape changes dramatically as the dimensionality is reduced. The illusion arises from our brains attempting to interpret the lower-dimensional representation.

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Applications in Simulation

Dimensional collapse is widely used in computer graphics for rendering complex 3D scenes efficiently. It’s a form of compression that reduces the amount of data needed to represent an object.

It's also utilized in areas like data visualization, where high-dimensional datasets are represented in lower dimensions for easier analysis and understanding.

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Frequently asked questions

Try it live

Everything above runs in your browser — open SPH Fluid and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.