Question: A regular tetrahedron has three of its vertices at $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. If all coordinates of the fourth vertex are integers, find its coordinates.

Sun, 03 May 2026 01:30:04 +0000

**Discover the Hidden Geometry Behind a Regular Tetrahedron — How Integer Coordinates Reveal a Hidden Symmetry** When exploring three-dimensional space, few shapes spark curiosity quite like the regular tetrahedron — a pyramid with equilateral triangular faces, where every edge is equal. A current question drawing quiet attention across tech, math, and design communities is: *A regular tetrahedron has three of its vertices at $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. If all coordinates of the fourth vertex are integers, find its coordinates.* This inquiry isn’t just academic — it reflects a broader fascination with precise spatial logic, symmetrical patterns, and integer-based structures that challenge intuitive geometry. Recent interest in this question mirrors growing curiosity in structural mathematics and crystallography, fields that underlie modern architecture, materials science, and even art. With increasing focus on precision in design and modeling — especially in digital workspaces — discovering exact coordinates that satisfy symmetry and distance rules offers tangible intellectual rewards. The challenge is deceptively simple: given three points in a familiar 3D lattice, can a fourth point exist with integer values that preserve uniform edge length, ensuring every side of the tetrahedron measures the same? ### Why This Question Resonates Across the US Community