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The $\infty$-Oreo$^{^\circledR}$

Vicente Bosca

Abstract

What happens when a food product contains a version of itself? The Oreo Loaded -- a cookie whose filling contains real Oreo cookie crumbs -- can be viewed as the result of mixing a Mega Stuf Oreo into a Mega Stuf Oreo. Iterating this process yields a sequence of increasingly self-referential cookies; taking the limit gives the $\infty$-Oreo. We model the iteration as an affine recurrence on the creme fraction of the filling, prove convergence, and compute the limit exactly: the stuf of the $\infty$-Oreo is approximately $95.8\%$~creme and $4.2\%$~wafer. We then extend the framework to pairs of foods that reference each other, deriving a coupled recursion whose fixed point defines a \emph{bi-$\infty$ food}, and illustrate the construction with M\&M Cookies and Crunchy Cookie M\&M's. Finally, we classify $\infty$-foods by the number of foods in the recursion and introduce \emph{homological foods}, whose recursive structure is governed by cycles in a directed graph of commercially available products. We close with a conjecture. All products used in this paper can be purchased at a supermarket.

The $\infty$-Oreo$^{^\circledR}$

Abstract

What happens when a food product contains a version of itself? The Oreo Loaded -- a cookie whose filling contains real Oreo cookie crumbs -- can be viewed as the result of mixing a Mega Stuf Oreo into a Mega Stuf Oreo. Iterating this process yields a sequence of increasingly self-referential cookies; taking the limit gives the -Oreo. We model the iteration as an affine recurrence on the creme fraction of the filling, prove convergence, and compute the limit exactly: the stuf of the -Oreo is approximately ~creme and ~wafer. We then extend the framework to pairs of foods that reference each other, deriving a coupled recursion whose fixed point defines a \emph{bi- food}, and illustrate the construction with M\&M Cookies and Crunchy Cookie M\&M's. Finally, we classify -foods by the number of foods in the recursion and introduce \emph{homological foods}, whose recursive structure is governed by cycles in a directed graph of commercially available products. We close with a conjecture. All products used in this paper can be purchased at a supermarket.

Paper Structure

This paper contains 19 sections, 5 theorems, 41 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Motivations
  3. Preliminaries---The Anatomy of an Oreo
  4. Components and notation
  5. The Oreo product line
  6. The $\infty$-Oreo
  7. The recursion
  8. The composition equations
  9. Solving the recursion
  10. Bi-$\infty$ Foods
  11. Coupled recursion
  12. The composition equations
  13. Solving the coupled recursion
  14. Classification of $\infty$-Foods
  15. Mono-$\infty$ foods
  16. ...and 4 more sections

Key Result

Lemma 1

The sequence $(w_n)$ defined by eq:w-recursion converges to the unique positive root of The limiting stuf fraction is $m^* = (m_s + w^*)/(m_s + m_w + w^*)$. $\blacktriangleleft$$\blacktriangleleft$

Figures (13)

  • Figure 1: Flavor innovation at three levels of abstraction.
  • Figure 2: Original Oreo broken down into its stuf fraction $s$ and wafer fraction $1-s$.
  • Figure 3: The Oreo product line as a sequence of enhancements.
  • Figure 4: The first three iterations.
  • Figure 5: The $\infty$-Oreo: the limit of the Oreo $n$-Loaded as $n \to \infty$.
  • ...and 8 more figures

Theorems & Definitions (35)

  • Definition 2.1: Wafer and Stuf
  • Definition 2.2: Stuf fraction
  • Definition 3.1: The Oreo $n$-Loaded
  • Definition 3.2: The $\infty$-Oreo
  • Definition 3.3: Stuf fraction
  • Remark 1
  • Definition 3.4: Loaded creme fraction
  • Definition 3.5: Creme fraction of the stuf
  • Definition 3.6: Base creme fraction
  • Lemma 1: Limiting stuf fraction
  • ...and 25 more