Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Computing the 4D Geode

Dean Rubine

TL;DR

The paper tackles the challenge of computing the four-dimensional Geode element G[1000,1000,1000,1000] within the hyper-Catalan framework. It builds on Wildberger and Rubine’s geometric polynomial structure, deploying a slice-based recurrence and ratio-driven hyper-Catalan calculations to manage combinatorial growth. The author implements a cache- and slice-oriented algorithm to extract a single Geode element efficiently, reporting a 6303-digit result and outlining the computational roadmap. The work demonstrates a scalable approach to higher-dimensional Geode computations and sketches pathways for extending to even higher dimensions, highlighting both the potential and the remaining challenges.

Abstract

The closed form for the hyper-Catalan number C[m2,m3,m4,...], which counts the number of subdivisions of a roofed polygon into m2 triangles, m3 quadrilaterals, m4 pentagons, etc., has been known since 1940. In 2025, Wildberger and Rubine showed its generating series S[t2,t3,t4,...] is a zero of the general geometric univariate polynomial. They note the factorization S=(t2 + t3 + t4 + ...)G, where the factor G is called the Geode. Later in 2025, Amderberhan, Kauers and Zeilberger issued a challenge to compute G[1000,1000,1000,1000], the coefficient of $t_2^{1000}t_3^{1000}t_4^{1000}t_5^{1000}$ in G. The reward is a donation to OEIS. We describe the computation, give the value and claim the reward.

Computing the 4D Geode

TL;DR

The paper tackles the challenge of computing the four-dimensional Geode element G[1000,1000,1000,1000] within the hyper-Catalan framework. It builds on Wildberger and Rubine’s geometric polynomial structure, deploying a slice-based recurrence and ratio-driven hyper-Catalan calculations to manage combinatorial growth. The author implements a cache- and slice-oriented algorithm to extract a single Geode element efficiently, reporting a 6303-digit result and outlining the computational roadmap. The work demonstrates a scalable approach to higher-dimensional Geode computations and sketches pathways for extending to even higher dimensions, highlighting both the potential and the remaining challenges.

Abstract

The closed form for the hyper-Catalan number C[m2,m3,m4,...], which counts the number of subdivisions of a roofed polygon into m2 triangles, m3 quadrilaterals, m4 pentagons, etc., has been known since 1940. In 2025, Wildberger and Rubine showed its generating series S[t2,t3,t4,...] is a zero of the general geometric univariate polynomial. They note the factorization S=(t2 + t3 + t4 + ...)G, where the factor G is called the Geode. Later in 2025, Amderberhan, Kauers and Zeilberger issued a challenge to compute G[1000,1000,1000,1000], the coefficient of in G. The reward is a donation to OEIS. We describe the computation, give the value and claim the reward.
Paper Structure (11 sections, 6 theorems, 14 equations, 10 figures)

This paper contains 11 sections, 6 theorems, 14 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The Geometric Polynomial Formula
  3. The Geode
  4. An OEIS Tangent
  5. The Geode Recurrence
  6. Caching
  7. Geode Slices for Computational Efficiency
  8. Hyper-Catalan Ratios
  9. Efficiently Calculating Hyper-Catalan Slices
  10. G[1000,1000,1000,1000]
  11. Future Directions

Key Result

Theorem 1

The number of subdigons of type ${\mathbf m}=[m_2, m_3, m_4, m_5]$ is where $E_{\mathbf m} \equiv 1+2m_2 + 3m_3 + 4m_4 + 5m_5$ is the number of edges in a subdigon of type ${\mathbf m}$, $V_{\mathbf m} \equiv 2+m_2 + 2m_3 + 3m_4 + 5m_5$ is the number of vertices in a subdigon of type ${\mathbf m}$, and ${\mathbf m}! \equiv m_2! \, m_3! \, m_4! \, m_5!$.

Figures (10)

  • Figure 1: A few of the $C[3,2,1]=43680$ subdigons subdivided into three triangles, two quadrilaterals and a pentagon.
  • Figure 2: Straightforward hyper-Catalan and Geode calculation
  • Figure 3: Calculation with cached Geode and factorials
  • Figure 4: The lattice of Geode elements used to compute $G[2,2,2,2]$; the number before the colon is the initial recursion depth.
  • Figure 5: Geode Slice Calculation
  • ...and 5 more figures

Theorems & Definitions (8)

  • Theorem 1: 4D Hyper-Catalan Closed Form, Erdélyi and Etherington, 1940
  • Theorem 2: The geometric quintic formula, Wildberger and Rubine, 2025
  • Theorem 3: The Geode factorization, Wildberger and Rubine, 2025
  • Theorem 4: The 4D hyper-Catalan / Geode sum
  • proof
  • Theorem 5: The 4D Geode Recurrence
  • Theorem 6: Ratio of hyper-Catalan neighbors
  • proof