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Squaring the Circle

Short answer: A sphere built from cubes, like a voxel model, can only approximate the curve: flat faces and square edges never match a continuous surface. Using more, smaller cubes makes the shape smoother, but some stair-stepping always remains.

The title echoes the classic problem of squaring the circle with compass and straightedge, which Ferdinand von Lindemann proved impossible in 1882. The footage is an animation of block spheres with ever smaller cubes.

VideoWrite-up2 sources
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What should you know about Squaring the Circle?

Quick facts. Each one stands alone and has a stable ID you can cite.

  1. Cubes have flat faces.

    DD-328-F01 · Write-up · Exhibit A (Flat faces)

  2. More cubes approximate a sphere better.

    DD-328-F02 · Write-up · Exhibit B (Approximation)

Video transcript: Squaring the Circle: No Number of Cubes Makes a Perfectly Smooth Sphere

Did you know, creating a perfectly smooth sphere from square blocks is mathematically impossible because square blocks have flat, straight edges, while a sphere has a continuous curved surface. No matter how many blocks are used, the flat surfaces of the block will never fully conform to the smooth, rounded surface of a true sphere. However, by increasing the number of smaller blocks, you can more closely approximate the shape of a sphere, with the curvature becoming smoother as more blocks are added. Despite this, some angularity will always remain, due to the inherent shape of the square blocks. NOW YOU KNOW!

Burned-in captions, read frame by frame (see transcript.md) · 28.1s Reel ·

Video coming soonSquaring the Circle · Instagram Reel
Squaring the Circle: No Number of Cubes Makes a Perfectly Smooth Sphere Instagram Reel · · 28.1s

The write-up

No number of cubes makes a perfectly smooth sphere. Smaller blocks get closer, but corners always remain.

  1. Exhibit A

    Flat faces

    Cubes have flat faces.

  2. Exhibit B

    Approximation

    More cubes approximate a sphere better.

  3. Exhibit C

    Never exact

    Corners always remain.

  4. Exhibit D

    Voxels

    3D pixels are called voxels.

  5. Exhibit E

    1882

    Squaring the circle was proved impossible in 1882.

Questions people ask about Squaring the Circle

What are 3D pixels called?

Volume elements.

Source: Write-up · Exhibit A (Flat faces) · Wikipedia: Voxel

Can cubes make a perfect sphere?

Only approximate.

Source: Write-up · Exhibit B (Approximation) · Wikipedia: Voxel

What helps the approximation?

More of them.

Source: Write-up · Exhibit C (Never exact) · Wikipedia: Voxel

When was squaring the circle proved impossible?

Lindemann.

Source: Write-up · Exhibit D (Voxels) · Wikipedia: Voxel

What always remains?

Stair steps.

Source: Write-up · Exhibit E (1882) · Wikipedia: Squaring the circle

Test Your Knowledge

All quizzes →

5 questions, all from this case file. Answers are revealed instantly.

  1. What are 3D pixels called?

    1. Voxels
    2. Pixels
    3. Atoms
    4. Tiles
    Show the answer

    Answer: A. Voxels Volume elements. Source: Write-up · Exhibit A (Flat faces) · Wikipedia: Voxel

  2. Can cubes make a perfect sphere?

    1. No
    2. Yes
    3. Only 100 cubes
    4. Only in space
    Show the answer

    Answer: A. No Only approximate. Source: Write-up · Exhibit B (Approximation) · Wikipedia: Voxel

  3. What helps the approximation?

    1. Smaller cubes
    2. Bigger cubes
    3. Paint
    4. Glue
    Show the answer

    Answer: A. Smaller cubes More of them. Source: Write-up · Exhibit C (Never exact) · Wikipedia: Voxel

  4. When was squaring the circle proved impossible?

    1. 1882
    2. 1066
    3. 1969
    4. 2001
    Show the answer

    Answer: A. 1882 Lindemann. Source: Write-up · Exhibit D (Voxels) · Wikipedia: Voxel

  5. What always remains?

    1. Some angularity
    2. Nothing
    3. Holes
    4. Color
    Show the answer

    Answer: A. Some angularity Stair steps. Source: Write-up · Exhibit E (1882) · Wikipedia: Squaring the circle

Cite this case file

Easterling, Marcus. "Squaring the Circle." The Detail Department (DetailEnjoyer), case file DD-328, last updated Sep 21, 2024. https://detailenjoyer.com/case/squaring-the-circle/

Permalink: https://detailenjoyer.com/case/squaring-the-circle/ · Last updated

Markdown version

Cite a single fact by its ID, for example DD-328-F01 → https://detailenjoyer.com/case/squaring-the-circle/#DD-328-F01