Arrangements of n Planes Resulting in One Bounded Tetrahedral Chamber

Author

• Ava KnightFollow

College

College of Engineering and Polymer Science

Date of Last Revision

2026-04-28 12:30:50

Major

Applied Mathematics

Honors Course

MATH498-001

Number of Credits

2

Degree Name

Bachelor of Science

Date of Expected Graduation

Spring 2026

Abstract

This paper investigates the combinatorial geometry of plane arrangements in three-dimensional space, focusing on configurations that produce exactly one bounded tetrahedral chamber. We define T(n) as the number of face-combinatorial equivalence classes of arrangements of n planes in ℝ³ containing exactly one bounded tetrahedral chamber. Known values — T(3) = 0, T(4) = 1, and T(5) = 2 — are established through direct construction, while T(6) remains an open problem. This paper contributes experimental evidence toward resolving T(6) by systematically extending the two valid 5-plane arrangements and verifying, through a plane removal argument, that each yields a valid plane configuration that has one bounded tetrahedral chamber or no bounded chamber. This produces at least two distinct equivalence classes for T(6), showing T(6) ≥ 2. Whether additional valid arrangements exist remains an open question. These findings highlight the combinatorial constraints that govern which plane arrangements can produce exactly one bounded tetrahedral chamber, and motivate the broader questions of whether T(n) admits a general formula and how structural complexity scales with n.

Research Sponsor

Stefan Forcey

First Reader

James P Cossey

Second Reader

Andreas Aristotelous

Honors Faculty Advisor

James P Cossey

Proprietary and/or Confidential Information

No

Community Engaged Scholarship

No

Recommended Citation

Knight, Ava, "Arrangements of n Planes Resulting in One Bounded Tetrahedral Chamber" (2026). Williams Honors College, Honors Research Projects. 2097.
https://ideaexchange.uakron.edu/honors_research_projects/2097