This app was inspired by the book Flatland by Edwin A. Abbott. It is about a society of flat shapes: triangles, squares, hexagons etc., who live in a horizontal two-dimensional plane called Flatland. They can only move and see within their plane; they know what north, south, east and west mean, but they have no conception of up or down. The narrator of the story is a Square, who is visited by a Cube* one day. The Square does not understand what a cube is. In the book, the Square explains to the Cube how their society works, and the Cube tries to explain to the Square what the third dimension is.

To show himself to the Square, the Cube first moves up and down through Flatland face-first. What the Square sees is another square (the horizontal intersection of the Cube with Flatland) suddenly appearing out of nowhere, then staying put for a while, and then disappearing again. Next, the Cube rotates itself and moves up and down edge-first. Now the Square sees a line appearing out of nowhere, which turns into a long narrow rectangle, which gets wider and wider for a while, then it gets narrower and narrower again, until it turns back into a line and then it disappears. Finally, the Cube rotates itself once more, and moves up and down vertex-first. Now the Square sees a point appearing out of nowhere, which turns into a small triangle, which gets larger and larger for a while, then its vertices get cut off and it turns into a hexagon. When the Cube is exactly half way through, the Square can see the Cube's horizontal intersection with Flatland as a regular hexagon. As the Cube moves further, the hexagon turns back into a triangle, which then gets smaller and smaller, and finally the triangle turns into a point and disappears.

This app does the same thing one dimension higher. Instead of a Cube visiting people who live in a two-dimensional plane, it shows a Hypercube (four-dimensional cube) visiting people, like you and me, who live in a three-dimensional space.

When the app starts, the Hypercube is sitting face-first exactly half way through our three-dimensional space. We can see the "horizontal" intersection of the Hypercube with our space, which, as you have probably guessed, is a three-dimensional cube.

You can move the cube around in our space by dragging it with your fingers. It has six colored faces, which are the intersections of our space with six of the eight colored faces of the Hypercube. Each face of the Hypercube has a different color.

You can move the Hypercube "up" and "down" in the direction of the fourth dimension using the red slider. This direction is perpendicular to all our three coordinate axes x, y and z, and is just as difficult for us to imagine as our up and down are to the people of Flatland.

To make more interesting shapes, you can rotate the Hypercube using the three blue sliders. These sliders rotate the Hypercube around the pairs of axes xy, xz and yz, respectively. It is not hard to see that as you can rotate a cube in three-dimensional space around any one axis, you can rotate a hypercube in four-dimensional space around any pair of axes.

Try to set the blue sliders to make the Hypercube move through our space two-dimensional-face-first, edge-first, and vertex-first! This takes some thinking, but it's not difficult. Then move the Hypercube "up" and "down" using the red slider, and see how the intersection of the Hypercube with our three-dimensional space changes. What is the intersection exactly half way through in each of these three directions?

What is the most interesting shape you can make? What is the largest possible number of faces? What is the largest possible number of vertices?

Hypercube Viewer is free software. You can browse and download the source code at https://github.com/fgerlits/hypercube

* in the book, it's a Sphere, but spheres are boring