# Diagrammatics in Art and Mathematics

## Abstract

**:**

## 1. Introduction

## 2. Modeling the Universe: Tessellations

“Much as the discovery of these strange forms may be calculated to excite our curiosity, and to awaken an intense desire to learn something of the laws which give order to these wonderful systems, as yet, I think, we have no fair ground even for plausible conjecture.”

^{−13}, 7500, 5000, 45000000, 50000) described in paper [5]. Left photo on Figure 2 is obtained by NASA and The Hubble Heritage Team (STScI/AURA) in March 1997 and September 2000, telescope: Hubble Wide Field Planetary Camera 2.

“The assumption that the sum of three angles is less than ${180}^{\circ}$ leads to some curious geometry, quite different than ours, but thoroughly consistent.”

- Mathematics: determine the fundamental domain (region) i.e., the smallest region in the tessellation which can be used to recreate the tessellation by applying the symmetries of the tessellation.
- Art: choose a pattern to insert into the fundamental domain, and map it onto the plane via symmetries of the tessellation, along with the fundamental domain, and observe the pattern it forms.

## 3. Knot Theory: Knotting Mathematics and Art

## 4. Diagrammatic Categorification

- diagrams of knots and links for Khovanov link homology and the Knot Floer homology,
- graphs for the chromatic graph cohomology and the categorification of the Tutte polynomial.

“L’image est écrite et l’écriture forme des images... on peut dire qu’il y a une écriture, une graphologie dans toute image de même que dans toute écriture se trouve une image.”

## Acknowledgements

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**Figure 3.**M.C. Escher Circle Limit IV (Heaven and Hell), 1960, Woodcut Printed from Two Blocks and Angels and Demons.

**Figure 4.**Spherical or elliptic (3,3,3,3,5), Euclidean (3,3,3,3,6) and hyperbolic (3,3,3,3,7) tessellations.

**Figure 5.**Escher-like tessellation determined by symbol (6,6,6,6), superimposed with the wire model of the basic tessellation on the right.

**Figure 6.**Hyperbolic twittering machine tessellation and its variations, by R. Sazdanovic and M. Sremcevic, 2000 [9].

**Figure 13.**Link $101\ast \mathrm{21.210.21.210.21.210.21.210.21.210}$ by S.Jablan, created using LinKnot and Knotplot.

**Figure 15.**Zeroes of the Jones polynomials of pretzel knots and links with at most 25 crossings by S.Jablan, created using LinKnot and Knotplot.

**Figure 16.**Plots of zeroes of the chromatic and the Jones polynomial by R. Sazdanovic using LinKnot and Polynomioraphy by B. Kalantari.

**Figure 18.**Categorification of quantum groups and the Casimir element: notebook and blackboard by A. Lauda.

**Figure 19.**Diagrams of basis elements in projective, standard, and simple modules in categorification of the Hermite polynomials, respectively.

© 2012 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Sazdanovic, R.
Diagrammatics in Art and Mathematics. *Symmetry* **2012**, *4*, 285-301.
https://doi.org/10.3390/sym4020285

**AMA Style**

Sazdanovic R.
Diagrammatics in Art and Mathematics. *Symmetry*. 2012; 4(2):285-301.
https://doi.org/10.3390/sym4020285

**Chicago/Turabian Style**

Sazdanovic, Radmila.
2012. "Diagrammatics in Art and Mathematics" *Symmetry* 4, no. 2: 285-301.
https://doi.org/10.3390/sym4020285