Geometry in Other Dimensions

Pictured: A simulated hyperbolic geometry in which parallel lines diverge, so that the higher one goes, the more space there is. The shapes that are stacked on top of each other in this image are completely cubic and congruent.
Source: Portals to Non-Euclidean Geometries, ZenoRogue on YouTube (2022).

Important note

Introduction

You've learned about regular geometry, on a three-dimensional plane of existence (our world) — things like the Pythagorean Theorem and whatnot. But what happens when the number of dimensions changes? Or when the dimensions aren't plane-shaped anymore, but spheres or saddles? These concepts have been one of physicists' fascinations since the 1800s.

This website is meant to provide a brief insight into all that non-Euclidean geometry has to offer. See the Conventional Euclidean Rules tab to look at the history of "normal" (Euclidean) geometry.

Navigation

1. Conventional Euclidean Rules - explanation of how we understand Euclidean geometry, and how people began to think about non-Euclidean geometry
2. Extradimensional Lenses - explores ways to look at and think about creatures and objects of other dimensions
3. Notable Experiments - lists several interesting geometry-related experiments and impossible objects for which further reading is recommended
4. The Shape of Space - gravity and different shapes of existence
5. Forced Perspective - not optical illusions; this is about our inability to perceive a fourth or fifth or sixth dimension of existence and why it's impossible to visualize
6. Credits - about, references, & bibliography