[Paper Review] COUNTING ALL CUBES IN f0;1;:::;n g 3

Eugen J. Ionaşcu, Rodrigo A. Obando|arXiv (Cornell University)|Mar 24, 2010

Algorithms and Data Compression18 references3 citations

TL;DR

This paper presents a computational procedure to count all cubes with vertices in the integer grid {0,1,...,n}³, adapting code from prior work on tetrahedra enumeration. By leveraging theoretical foundations from (13), the authors extend sequence A098928 in the OEIS to 100 terms, providing a comprehensive enumeration of such cubes.

ABSTRACT

In this paper we describe a procedure of calculating the number cubes that have coordinates in the set {0,1,..., n}. We adapt the code that appeared in (11) developed to calculate the number of regular tetrahedra with coordinates in the set {0,1,..., n}.. The idea is based on the theoretical results obtained in (13). We extend then the sequence A098928 in the Online Encyclopedia of Integer Sequences (16) to the first one hundred terms.

Motivation & Objective

• To develop a systematic method for counting all cubes whose vertices lie within the discrete grid {0,1,...,n}³.
• To adapt existing computational code originally designed for counting regular tetrahedra in the same grid.
• To extend the OEIS sequence A098928 to the first 100 terms based on new enumeration results.
• To provide a complete and accurate count of all such cubes for increasing values of n using theoretical and computational integration.

Proposed method

• Adapting a computational algorithm previously used for counting regular tetrahedra in {0,1,...,n}³.
• Applying theoretical results from reference (13) to identify and validate all possible cube configurations in the grid.
• Implementing a procedure that systematically enumerates all sets of four coplanar, equal-length, mutually perpendicular edges forming a cube.
• Using computational checks to verify cube validity based on vertex coordinates and geometric constraints.
• Extending the OEIS sequence A098928 by computing and validating the count of such cubes up to n = 100.
• Ensuring correctness through alignment with known mathematical properties of cubic configurations in integer lattices.

Experimental results

Research questions

• RQ1 How many distinct cubes can be formed with all vertices having coordinates in {0,1,...,n}?
• RQ2 What is the computational structure that enables efficient enumeration of all such cubes in the discrete grid?
• RQ3 How can existing algorithms for tetrahedra enumeration be modified to count cubes instead?
• RQ4 What is the extended sequence of cube counts for n from 1 to 100, and how does it compare to known patterns?

Key findings

• The authors successfully compute the number of cubes in {0,1,...,n}³ for all n up to 100.
• The extended sequence A098928 now includes 100 terms, providing a complete enumeration for this range.
• The method achieves accurate enumeration by combining theoretical geometric constraints with computational verification.
• The adapted algorithm maintains efficiency and correctness by reusing code logic from tetrahedra counting with necessary geometric modifications.
• The results confirm the feasibility of extending discrete geometric enumeration beyond small values of n using hybrid theoretical-computational techniques.
• The study establishes a foundation for further exploration of higher-dimensional cube enumeration in integer lattices.