# Squaring the Circle (DD-328)

> 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.

- Department: Science
- First published: 2024-09-21
- Last updated: 2024-09-21
- Permalink: https://detailenjoyer.com/case/squaring-the-circle/
- Author: Marcus Easterling (DetailEnjoyer), The Detail Department

## On the timeline

- **1882**: Lindemann proves squaring the circle impossible

Full timeline: https://detailenjoyer.com/timeline/?case=squaring-the-circle

## Quick facts

- **DD-328-F01**: Cubes have flat faces. _(Write-up · Exhibit A (Flat faces))_
- **DD-328-F02**: More cubes approximate a sphere better. _(Write-up · Exhibit B (Approximation))_

## Write-up

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

1. **Flat faces**: Cubes have flat faces.
2. **Approximation**: More cubes approximate a sphere better.
3. **Never exact**: Corners always remain.
4. **Voxels**: 3D pixels are called voxels.
5. **1882**: Squaring the circle was proved impossible in 1882.

## Reel vs. record

- **Note**: None needed.

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

- Duration: 28.1 s · Published: 2024-09-21
- File: https://detailenjoyer.com/assets/videos/DANCeoLJIHU.mp4

### Transcript

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!

## Questions & answers

**Q: What are 3D pixels called?**  
A: Volume elements. _(Source: Write-up · Exhibit A (Flat faces) · Wikipedia: Voxel)_

**Q: Can cubes make a perfect sphere?**  
A: Only approximate. _(Source: Write-up · Exhibit B (Approximation) · Wikipedia: Voxel)_

**Q: What helps the approximation?**  
A: More of them. _(Source: Write-up · Exhibit C (Never exact) · Wikipedia: Voxel)_

**Q: When was squaring the circle proved impossible?**  
A: Lindemann. _(Source: Write-up · Exhibit D (Voxels) · Wikipedia: Voxel)_

**Q: What always remains?**  
A: Stair steps. _(Source: Write-up · Exhibit E (1882) · Wikipedia: Squaring the circle)_

## Sources

- [Wikipedia: Voxel](https://en.wikipedia.org/wiki/Voxel)
- [Wikipedia: Squaring the circle](https://en.wikipedia.org/wiki/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/