ncube - A Generalized Hypercube Visualizer

ncube allows you to visualize rotating hypercubes of arbitrary dimensions. It works by rotating the hyperdimensional vertices and applying a chain of perspective projections to them until the 3rd dimension is reached. Everything is generated in real time just from the dimension number.

On December 13th 2023, ncube made it to #4 of Hacker News (post).

• Features
• But what am I actually visualizing?
• Web
• Installation
  • Download the pre-built binaries
  • Install from crates.io
  • Install from source
  • Uninstall
• Contributing

Features

• Real time control of the simulation, such as tweaking the angular velocity factor of any plane of rotation
• Exporting and loading custom configurations as files

But what am I actually visualizing?

Let's use the 7-cube demo GIF above as an example. In that specific case, you are looking at:

• A 3 dimensional perspective projection of...
  • a 4 dimensional perspective projection of...
    • a 5 dimensional perspective projection of...
      • a 6 dimensional perspective projection of...
        • a 7 dimensional hypercube undergoing a rotation about the q1q4 (X-W1) and q2q3 (Y-Z) orthogonal planes.

Okay, let's unpack this... We, simple 3D creatures in a (at least apparently) 3D Universe can easily perceive 3D objects and it gets even easier for those of smaller dimensions, a polygon, a line, a point... But what if we want to see beyond?

Well, imagine you're a 2D being, you live in a flat universe, in a plane, you see in 1D (a line), then you try to visualize a 3D object, so your 3D friends intersect a cube into your line of sight, but you only see an infinitesimal slice of it, you can never see it fully.

The exact same problem applies to us when we try to comprehend a 4 or higher dimensional object. So how can we solve this problem?

We need to represent the object in lesser dimensions. And we see such representations everyday... Shadows. Shine a light through the cube and you'll end up with all its features compressed into 2 dimensions, surely it's not the same as the cube itself, but it's a lot better than the thin slice! And believe it or not that's what the math of this project is all about, shadows... Another word for those is perspective projections.

We cast a shadow, and then we cast a shadow of a shadow and so on, with each step peeling away one dimension until we end up with something 3D that we can see and examine fully. Having that, we can perform rotations on the hypercube and see how its projection changes.

There, we have weirder kinds of rotations that we don't have in 3D space, called double, triple, and so on, which occur when you have multiple simple rotations at the same time in a way that don't interfere with each other. For example in 3D space you can rotate along the X-Y plane, but if you go 4D you have not 1 but 2 dimensions free in which you can perform another rotation.

Web

A web version for this app is available at ncube.ndavd.com via the WASM build.

Installation

Download the pre-built binaries

Pre-built binaries for Windows, Linux, MacOS and WASM can be found in the releases page.

Install from crates.io

Install cargo and run the install command:

cargo install ncube

Install from source

Install cargo, clone the repository and run the install command:

git clone https://github.com/ndavd/ncube
cd ncube
cargo install --path .

Uninstall

cargo uninstall ncube

Contributing

Contributions are very welcome! Those being pull requests, issues or even ideas.

• If you have an idea or question feel free to create a discussion in the discussions page
• If you'd like to showcase your findings there's community showcase