A 1.58-Dimensional Object

Source: A 1.58-Dimensional Object - Numberphile, Numberphile, 20:25, uploaded 2025-03-07, playlist index 314.

Ben Sparks opens by asking whether anyone has seen a 1.58-dimensional shape. The question grows out of an older Numberphile video about the Chaos Game, where a random sequence of simple moves produces a Sierpiński triangle. The new video follows that game into fractal dimension, then into three-dimensional objects that can look like squares from selected directions.

From the Chaos Game to the Sierpiński triangle

The game starts with three points at the corners of a triangle and a fourth point somewhere inside it. A die or random choice selects one corner, and the moving point travels halfway towards that corner. The process repeats. A few hundred hand-drawn moves are tedious. A computer can keep going until the pattern emerges.

Sparks then removes the randomness to make the structure easier to see. Each point branches towards all three corners, so every point becomes three points at the next level. The number of points grows exponentially, yet the same pattern appears. Repeating the operation leaves the original points behind and replaces each one with three smaller copies. The result is the Sierpiński triangle, also called the Sierpiński gasket.

The interactive construction matters because Sparks can change the inputs and watch the result move. The three points can form a crooked triangle rather than an equilateral one. The game can use four or five points, and the moving point can travel a third or two-thirds of the way towards its target instead of travelling halfway. These changes produce different patterns. The point of the demonstration is the rule that produces them, rather than the appearance of one especially pretty image.

Dimension as a rule for scaling

Sparks introduces dimension through a familiar scaling exercise. A square made twice as large needs four copies of the original square. A square made three times as large needs nine copies. The number of copies follows the square of the scale factor, which is the familiar two-dimensional rule.

A cube made twice as large needs eight copies, so its copy count follows the cube of the scale factor. A line made three times as large needs three copies, which gives the one-dimensional case. In these examples, the dimension is the exponent in the relation between the scale and the number of copies.

The Sierpiński triangle follows the same calculation. A copy made twice as large needs three smaller copies because the central opening leaves room for three corners. The exponent therefore solves

Taking logarithms gives

The number describes how the shape scales. It does not claim that the object occupies an ordinary physical space with 1.58 coordinate axes. Sparks presents this as one technical definition of dimension, close to the Hausdorff-dimension language used for fractals, and mentions Minkowski dimension as another notion mathematicians use. The video leaves the relationships between those definitions aside.

More holes, a higher dimension

The Sierpiński carpet applies the same reasoning to a square grid. The next repeatable scale is three times the size, and the pattern needs eight copies because the central square is missing. Its dimension solves

which gives

Both shapes contain holes at every scale. The carpet feels closer to an area because its dimension sits nearer to two. The triangle feels closer to a line because its value sits nearer to one. The scale calculation gives a number for that difference even though both objects remain full of openings.

Sparks briefly raises the question of whether every number can occur as a dimension, then leaves it open. A related thought experiment asks what it would mean to live in a 1.58-dimensional world, as in Edwin Abbott Abbott’s Flatland. His answer is that the idea is confusing enough to postpone. The video stays with consequences that the scaling rule can show directly.

The same numbers inside the Chaos Game

The Chaos Game makes the copy count visible. The Sierpiński triangle uses three target points and a halfway move. To build the carpet, Sparks places eight target points at the positions of the eight scaled copies and moves two-thirds of the way towards each one. The number of points and the jump ratio now describe the same self-similar structure that produced the dimension calculation.

This also makes room for a useful failure. Four points arranged as a square seem like a natural extension of the three-point triangle. Keeping the halfway move produces a blank-looking field. Sparks says that the Chaos Game page on Wikipedia claims this arrangement does not produce a fractal. He had checked the page shortly before filming and disagrees with that conclusion. When he drags one point away from the square, four copies become visible and the pattern starts to look three-dimensional.

That small experiment changes the problem. The points feel as if they sit on the corners of a pyramid, so Sparks moves the game into three dimensions. Four points arranged as a tetrahedron generate a Sierpiński tetrahedron. A regular tetrahedron makes the structure clear, and a 3D-printed version gives the calculation something that can sit on a table.

A tetrahedron with dimension two

The printed fractal tetrahedron scales by a factor of two and needs four copies. Its dimension therefore solves

so $d=2$. The object exists in three-dimensional space and has a fractal dimension of two. That result feels wrong when the object is held in the hand because its holes keep it from behaving like a solid volume.

The projection explains part of the tension. From the right direction, the tetrahedron’s outline becomes a square. Sparks lines up the object with the camera and finds three different viewing directions that produce a square projection. A two-dimensional area appears inside a three-dimensional object when the object is viewed along those directions. The dimension calculation records a property of the object’s copies and projections, rather than merely counting the dimensions of the room around it.

Sparks says Mandelbrot coined the word fractal to describe shapes with fractional dimensions. The tetrahedron shows why that name is suggestive without being a complete definition. Its dimension is the integer two, yet its repeated holes give it the same unsettled quality as the fractional examples. A Menger sponge supplies the other side of the range in the video. Its dimension lies between two and three because it fills more than an area whilst leaving holes throughout its structure.

The conclusion arrives through an accident of viewing. Sparks already knew the tetrahedron had dimension two, and he knew that a projection could look square. While dragging the model, he first sees a square and then realises that the apparent square comes from looking straight at the tetrahedron’s structure. The three square-producing directions belong to the same geometric object.

Hideki Tsuiki’s imaginary cubes

The video then turns to sculptures by Hideki Tsuiki. A tetrahedron fits inside a cube, and its projections look square from the relevant directions. Tsuiki calls this an “imaginary cube” because it has the cube’s square view from every direction shown in the demonstration whilst its actual geometry is a different shape.

Tsuiki has made more complicated examples. The T fractal produces square views along three directions. The H fractal is built from a hexagonal bipyramid and produces six. The extra footage after the main discussion shows another fractal built from a triangular antiprism. Smaller copies can be inserted into it without changing the square-view property, which gives the object its fractal form.

Tsuiki has published work on the directions from which these shapes show a surface with the desired projection. The video leaves those constructions at the level of an invitation to look further. It uses the models to show that a shape can carry an area-like projection whilst its three-dimensional construction remains full of holes.

Limits

The video is a visual explanation built around exact self-similar constructions. The logarithms support the dimensions of the examples under the scale-based rule, while the discussion of Hausdorff and Minkowski dimension remains introductory. It does not establish whether every real number can occur as a fractal dimension, develop the competing definitions, or prove the claims about all of Tsuiki’s sculptures.

The Wikipedia remark belongs to Sparks’s account of a page he read before filming. The note keeps it as part of the story because the apparent failure prompts the move into three dimensions. The description links Tsuiki’s site, related Numberphile videos, and a GeoGebra construction, but the note does not treat those links as independently read sources.

Further reading / references

• Ben Sparks, whose work and further Numberphile appearances are linked in the video description.
• Fractal Tetrahedron in GeoGebra, a companion video in which Sparks builds the object.
• The Chaos Game, the earlier Numberphile video revisited at the start.
• Hideki Tsuiki’s imaginary-cube work, linked by Numberphile for the sculptures and their geometry.