Question: A regular tetrahedron has three vertices at $(1, 1, 1)$, $(1, -1, -1)$, and $(-1, 1, -1)$. Find the integer coordinates of the fourth vertex.

Thu, 07 May 2026 00:26:50 +0000

**Why the Shape Question Has Surged in US STEM and Geometry Discussions** In recent years, unexpected geometric puzzles have sparked keen curiosity across digital communities, particularly in mobile searches tied to math, design, and spatial reasoning. A recurring query—how to determine the fourth integer coordinate for a regular tetrahedron given three known vertices—has gained significant traction in the United States. This happens against a backdrop of growing interest in 3D modeling, cryptography, event architecture, and educational tools that hinge on precise spatial understanding. Understanding the rules of symmetry and distance in regular polyhedra isn’t just academic—it influences design algorithms, structural engineering, and even artistic visualization. The specific problem of locating the missing vertex of a regular tetrahedron through integer coordinates may seem abstract, but it reveals foundational principles with practical applications in tech and academia. **Understanding the Tetrahedron’s Symmetry from These Points** The given vertices are $(1, 1, 1)$, $(1, -1, -1)$, and $(-1, 1, -1)$. To find the integer fourth vertex of a regular tetrahedron, we must calculate a point equidistant from all three given points with side length consistent across every edge. A regular tetrahedron requires all six edges to be equal in length. Calculating pairwise distances confirms these three points form an equilateral triangular face in 3D space. The missing point must preserve equal distances to each vertex while holding integer coordinates—a challenge that combines geometry and number theory.