Hinton's Tesseract, Hypercube in Four Axes

Tap or click to cycle through four rotation modes. The depth-cued lines encode the fourth dimension: edges near the w-axis extremes fade, edges near the projection center intensify. What reads as foreshortening is the shadow of a rotation you cannot directly see.

Mathematicians had been working fluently in four-dimensional space for decades before anyone had a reliable method for drawing a four-dimensional object on a two-dimensional page. The number of vertices, edges, faces, and cells in a tesseract (16, 32, 24, 8) follows directly from extending the combinatorial pattern of lower-dimensional cubes. What resisted that extension was the pictorial problem: how to make those counts visible without collapsing them to an abstract table.

Victor Schlegel's 1886 solution was projection with a carefully chosen viewpoint. Place the camera just outside one of the cubic cells, looking inward: the near cell expands to fill the frame, the far cell shrinks to a smaller cube nested inside the first, and the twelve remaining edges connect corresponding vertices of the two cubes. The result is a flat diagram that preserves the full combinatorial structure of the tesseract without distorting any edges into invisibility. Schlegel diagrams became the standard visualization tool for four-dimensional polytopes and remain so today, appearing in undergraduate topology textbooks in essentially the form he introduced them.

Charles Howard Hinton gave the tesseract its name in his 1888 book A New Era of Thought, deriving it from the Greek tessara (four) for the four edges meeting at each vertex. Hinton worked as a speculative thinker, not a research mathematician, and his project was pedagogical as much as mathematical: he believed that training spatial memory of four-dimensional structures could sharpen the mind in ways ordinary geometry could not. To that end he invented physical wooden models, color-coded cubes assembled and disassembled in sequences meant to build 4D intuition, and sold them. The models were unwieldy and his theoretical framework was eccentric. The name stuck, and the project of making hyperspace intuitive outlasted him.

When Manfred Mohr began drawing hypercubes in 1973, he was extending a systematic program he had been developing since 1969: use the computer to exhaustively explore a geometric structure, generating every configuration the structure permits and plotting them on paper. His early work used the cube as its fixed system. By the early 1970s he had extended the program to higher dimensions. The P-197 series (1977, exhibited at Galerie Weiller, Paris) applied the same logic to the hypercube: write an algorithm that generates projections of a four-dimensional cube, divide the cube's vertices into two groups, rotate each group independently, and plot the result as a plotter drawing on paper.

What separated Mohr's approach from mathematical illustration was the decision to let the algorithm run without selecting particularly beautiful outputs. The P-197 series is typically presented as a grid of 64 small images, every configuration the system produces. The visual interest comes from the full range: some projections are open and readable, some are tangled, some nearly degenerate. Mohr's concern was the complete space of hypercube drawings, a commitment that aligns him with composers like John Cage and movements like Fluxus more than with conventional mathematical visualization.

The artifact here uses two simultaneous rotation planes, which is the natural way to rotate a four-dimensional object. A 3D rotation has one plane of rotation and one fixed axis. A 4D rotation has two independent planes of rotation and no fixed axis; every point in the space is either moving or at the intersection of the two planes. The four modes cycle through different plane pairings: XZ/YW, XW/YZ, XZ/XW (coupled, sharing an axis), and YW/ZW (both involving the w-axis). The core of the rotation step is:

function rot4(v, plane, angle) {
  const [x, y, z, w] = v;
  const c = Math.cos(angle), s = Math.sin(angle);
  switch(plane) {
    case 'XZ': return [ x*c - z*s, y, x*s + z*c, w];
    case 'XW': return [ x*c - w*s, y, z, x*s + w*c];
    case 'YW': return [ x, y*c - w*s, z, y*s + w*c];
    case 'YZ': return [ x, y*c - z*s, y*s + z*c, w];
    // ...
  }
}

Each rotation leaves two coordinates fixed and rotates within the named plane. The two rotations are applied sequentially each frame, then the result is projected to 3D by perspective-dividing the w-axis (scale = fov / (fov + w)), and then projected again to 2D by discarding the z-axis after a camera-distance correction. Both projections are perspective rather than orthographic, which is why near edges appear larger than far ones and the depth-cueing records the projection arithmetic directly rather than serving as decoration.

Schlegel's 1886 diagram and Mohr's 1977 plotter series arrived through separate lines of development, in different disciplines and different countries, driven by different questions. Both converge on the same problem: how to represent an object with more dimensions than the surface you are drawing on. Every solution is a projection, and every projection involves a choice about what to preserve and what to collapse. Schlegel preserved combinatorial structure. Mohr preserved temporal variation across the full rotation space. The artifact here tries to hold both: the edges stay connected through the rotation, and the rotation continues through the space of views the four modes define, without settling at a canonical pose.