# GomJau-Hogg’s Notation for Automatic Generation of k-Uniform Tessellations with ANTWERP v3.0

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Semi-regular tilings (Archimedean or uniform) are polymorphic (several polygon types) and also vertex-transitive (Figure 2). There are only 8 of them.
- Demi-regular tilings (k-uniform) like semi-regular tilings are polymorphic but are not vertex-transitive (Figure 3). For instance, there are 20 2-uniform tessellations and there are 61 3-uniform (22 are 2-vertex and 39 are 3-vertex types).

^{6}; 3

^{6}; 3

^{4}.6, tells us there are 3 vertices with 2 different vertex types, so this tiling would be classed as a ‘3-uniform (2-vertex types)’ tiling. Broken down, 3

^{6}; 3

^{6}(both of different transitivity class), or (3

^{6})

^{2}, tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles). With a final vertex 3

^{4}.6, 4 more contiguous equilateral triangles and a single regular hexagon.

## 2. Problems of Cundy & Rollett Notation

^{6})

^{2}; 3

^{4}.6: If a single vertex was placed, surrounded by 4 triangles and a hexagon (3

^{4}.6), there would be 3 other vertices with 2 triangles (A, B and C in Figure 4c). From here, either the vertex type of 3

^{6}(as shown if Figure 4b) or 3

^{4}.6 (Figure 4c) is possible, and the notation gives no indication to which is correct.

## 3. GomJau-Hogg’s Notation: A New Notation

^{6})

^{2}; 3

^{4}.6 (the first of the three variants) is translated to 6-3-3/m30/r(v3). The stages are represented as blocks separated by a forward-slash (/). When split up, the very first block is the “Shape placement” stage, which takes care of placing the first regular polygons on the plane. The blocks after this are the transformation functions, of which there will be two or more.

#### 3.1. Stage 1: Polygon Placement

- Phase 1 (seed polygon phase): The very first phase will always contain a single number of either 3, 4, 6, 8 or 12. This is because there are seventeen combinations of regular polygons whose internal angles add up to 360°, however only eleven of these can occur in regular polygon tilings [4]. This defines the ‘seed polygon’, which is the first shape to be placed at the origin of the area to be covered.
- The seed polygon is always (except for the 3-sided polygon, equilateral triangle) placed at the origin of the plane so that the two sides that intersect the horizontal axis “x”, stay perpendicular to that axis (Figure 6). For an equilateral triangle the left-hand edge will be the one perpendicular to the x axis and will be aligned with the vertical axis ‘y’ [3] (Figure 7).
- Phase 2: Following the first phase, regular polygons are systematically placed clockwise around the available sides of the seed polygon, using 0 to skip a side of a polygon. The principle is to first fill the upper right quadrant and then move clockwise.
- Phase 3 (and 4, 5, etc.): Regular polygons are systematically placed clockwise around the available sides of the polygons placed in the previous phase, in the same manner as in phase 2.

^{6})

^{2}; 3

^{4}.6 in its former notation, which would become GomJau-Hogg’s notation 6-3-3/m30/r(v3) (Figure 6). With this new notation as shown above, the polygon placement stage (6-3-3) consists of:

- A seed polygon with 6 sides. In Figure 6, hexagon in dark blue.
- A following phase with a three-sided shape; placed on the first side clockwise of the y axis. In Figure 6, equilateral triangle in light blue.
- Followed by a final phase of one triangle; placed on the first available side clockwise, of the previously placed triangle. In Figure 6, equilateral triangle in white.

#### 3.2. Stage 2: First Transformation Function

- Type of transformation: It is represented in the notation by a single character. An ‘m’ (mirror) applies a reflection transformation and a ‘r’ applies a rotation transformation.
- Angle: When we have the polygons of the first stage, it is necessary to either rotate or reflect them by a specified angle. When no angle is specified it defaults to 180°.
- Origin: The origin of the transformation is specified between parentheses. When no origin is specified it defaults to the center of the coordinate system. There are 2 types of transformation origins, explained in the following lines.

#### 3.2.1. Origin 1. Center of the Coordinate System—Continuous Centered Transformation

- m30: Type of transform: m for mirroring. Angle 30°. An origin is not specified between parentheses, therefore the origin is the center of the coordinate system. Therefore this transform is going to reflect by 30° the elements of the previous phase obtained in Figure 6 along an axis passing by the center of the coordinate system.
- Then, it reflects 60° (30° × 2) the result of the previous transformation.
- Reflect 120° (60° × 2) the result of the previous transformation.
- Reflect 240° (120° × 2) the result of the previous transformation. This is the last reflection as 240° × 2 is 480° and is above the 360° limit.

#### 3.2.2. Origin 2. Points on the Polygons—Eccentric Transformation

#### 3.3. Stage 3: Repeating the Transformations

## 4. ANTWERP, the Software

## 5. Results

- The notation of the tilings can be automatically generated without ambiguity
- There is no repetition on the names of the tilings, they are unique.

## 6. Conclusions and Further Research

#### Building an Exhaustive Tessellation Database

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Array of figures representing graphically all the tilings gathered in this table with both notations.

## References

- Otero, C. Diseño Geométrico de Cúpulas No Esféricas Aproximadas por Mallas Triangulares con un Número Mínimo de Longitudes de Barra. Ph.D. Thesis, University of Cantabria, Santander, Spain, 1990. [Google Scholar]
- Cundy, H.M.; Rollett, A.P. Mathematical Models; Tarquin Publications: Stradbroke, UK, 1981. [Google Scholar]
- Gomez-Jauregui, V.; Otero, C.; Arias, R.; Manchado, C. Generation and Nomenclature of Tessellations and Double-Layer Grids. J. Struct. Eng.
**2012**, 138, 843–852. [Google Scholar] [CrossRef] - Grünbaum, B.; Shephard, G.C. Tilings and Patterns; W. H. Freeman & Co.: New York, NY, USA, 1986. [Google Scholar]
- Hogg, H.; Gomez-Jauregui, V. Antwerp v3.0.0. 2019. Available online: https://antwerp.hogg.io (accessed on 17 October 2021).
- Galebach, B.L. Number of N-Uniform Tilings. 2003. Available online: https://www.probabilitysports.com/tilings.html (accessed on 30 November 2021).
- Khan, A.A.; Laghari, A.A.; Awan, S.A. Machine Learning in Computer Vision: A Review. EAI Endorsed Trans. Scalable Inf. Syst.
**2021**, 8. [Google Scholar] [CrossRef]

**Figure 3.**Demi-regular tilings, 3 examples: (

**a**) 3

^{6}; 3

^{2}.4.3.4, (

**b**) 3.4.6.4; 3

^{2}.4.3.4 and (

**c**) 3.4.6.4; 3.4

^{2}.6.

**Figure 4.**Vertices (

**a**) 3

^{6}in yellow, (

**b**) 3

^{6}in green and (

**c**) 3

^{4}.6 (in blue), being (

**a**,

**b**) of different transitivity class, along with the composition of one of the three different tessellations with the same vertex notation: (3

^{6})

^{2}; 3

^{4}.6.

**Figure 5.**Three different tessellations with the same Cundy & Rollett nomenclature: (

**a**) [(3

^{6})

^{2}; 3

^{4}.6]

^{1}; (

**b**) [(3

^{6})

^{2}; 3

^{4}.6]

^{2}and (

**c**) [(3

^{6})

^{2}; 3

^{4}.6]

^{3}. In green, vertices (3

^{6}), in blue, vertices 3

^{4}.6. Their configuration is different for the three of them.

**Figure 8.**Continuous centered transformation (Mirroring): Example of 6-3-3/m30 with center in the origin of coordinate system.

**Figure 9.**Continuous centered transformation (Rotation): Example of 4-3-4,3/r90 with center in the origin of coordinate system.

**Figure 10.**Eccentric transformation (Mirroring): Example of 3-6/m(h7) with center in point h7 (seventh halfway point).

**Figure 16.**Tree data structure representation of an invalid notation 3-12,6. Invalid due to the angle of the empty space between the hexagon and the dodecagon.

**Figure 17.**The same tessellation pattern but with different notations: (

**a**) 4-4,4-3,4-6/m90/r(c5)/r(v1) and (

**b**) 6-3,4-0,4,4-0,4/m90/m90(c5)/m(v1).

**Table 1.**Transformation of Cundy & Rollett’s notation to GomJau-Hogg’s notation, up to 3-Uniform (3 Vertex Types). All of these can be seen in the tilings explorer application of Antwerp https://antwerp.hogg.io/library (accessed on 29 October 2021).

CUNDY & ROLLETT | GOMJAU-HOGG |
---|---|

REGULAR | |

3^{6} | 3/m30/r(h2) |

6^{3} | 6/m30/r(h1) |

4^{4} | 4/m45/r(h1) |

UNIFORM | |

3.12^{2} | 12-3/m30/r(h3) |

3.4.6.4 | 6-4-3/m30/r(c2) |

4.6.12 | 12-6,4/m30/r(c2) |

(3.6)^{2} | 6-3-6/m30/r(v4) |

4.8^{2} | 8-4/m90/r(h4) |

3^{2}.4.3.4 | 4-3-3,4/r90/r(h2) |

3^{3}.4^{2} | 4-3/m90/r(h2) |

3^{4}.6 | 6-3-3/r60/r(h5) |

2 UNIFORM | |

3^{6}; 3^{2}.4.3.4 | 3-4-3/m30/r(c3) |

3.4.6.4; 3^{2}.4.3.4 | 6-4-3,3/m30/r(h1) |

3.4.6.5; 3^{3}.4^{2} | 6-4-3-3/m30/r(h5) |

3.4.6.4; 3.4^{2}.6 | 6-4-3,4-6/m30/r(c4) |

4.6.12; 3.4.6.4 | 12-4,6-3/m30/r(c3) |

3^{6}; 3^{2}.4.12 | 12-3,4-3/m30/r(c3) |

3.12^{2}; 3.4.3.12 | 12-0,3,3-0,4/m45/m(h1) |

3^{6}; 3^{2}.6^{2} | 3-6/m30/r(c2) |

[3^{6}; 3^{4}.6]^{1} | 6-3,3-3/m30/r(h1) |

[3^{6}; 3^{4}.6]^{2} | 6-3-3,3-3/r60/r(h8) |

3^{2}.6^{2}; 3^{4}.6 | 6-3/m90/r(h1) |

3.6.3.6; 3^{2}.6^{2} | 6-3,6/m90/r(h3) |

[3.4^{2}.6; 3.6.3.6]^{1} | 6-3,4-6-3,4-6,4/m90/r(c6) |

[3.4^{2}.6; 3.6.3.6]^{2} | 6-3,4/m90/r(h4) |

[3^{3}.4^{2}; 3^{2}.4.3.4]^{1} | 4-3,3-4,3/r90/m(h3) |

[3^{3}.4^{2}; 3^{2}.4.3.4]^{2} | 4-3,3,3-4,3/r(c2)/r(h13)/r(h45) |

[4^{4}; 3^{3}.4^{2}]^{1} | 4-3/m(h4)/m(h3)/r(h2) |

[4^{4}; 3^{3}.4^{2}]^{2} | 4-4-3-3/m90/r(h3) |

[3^{6}; 3^{3}.4^{2}]^{1} | 4-3,4-3,3/m90/r(h3) |

[3^{6}; 3^{3}.4^{2}]^{2} | 4-3-3-3/m90/r(h7)/r(h5) |

3-UNIFORM (2 VERTEX TYPES) | |

(3.4.6.4)^{2}; 3.4^{2}.6 | 6-4-3,4-6,3/m30/r(c2) |

[(3^{6})^{2}; 3^{4}.6]^{1} | 6-3-3/m30/r(v3) |

[(3^{6})^{2}; 3^{4}.6]^{2} | 6-3-3-3-0,3/m30/r(v2) |

[(3^{6})^{2}; 3^{4}.6]^{3} | 6-3-3,3-3-3-0,3/r60/r(h7) |

3^{6}; (3^{4}.6)^{2} | 3-3,3-6/m90/r(h6) |

3^{6}; (3^{2}.4.3.4)^{2} | 3-4-3,3/m30/m(h2) |

(3.4^{2}.6)^{2}; 3.6.3.6 | 4-6,4-4,3,3/m90/r(h4) |

[3.4^{2}.6; (3.6.3.6)^{2}]^{1} | 4-6,4-0,3,3/m/r(v1)/r(h25) |

[3.4^{2}.6; (3.6.3.6)^{2}]^{2} | 4-6,4-0,3,3/m90/r(v1) |

3^{2}.6^{2}; (3.6.3.6)^{2} | 6-3,0,3,3,3,3/r(h4)/r(v15)/r(v30) |

(3^{4}.6)^{2}; 3.6.3.6 | 6-3,3-0,3/r/r(v1)/r(h12) |

[3^{3}.4^{2}; (4^{4})^{2}]^{1} | 4-4-4-3/m90/r(h4) |

[3^{3}.4^{2}; (4^{4})^{2}]^{2} | 4-4-3/r(h6)/m(h5)/r(h3) |

[(3^{3}.4^{2})^{2}; 4^{4}]^{1} | 4-4-3-3-4/m90/r(h10)/r(c3) |

[(3^{3}.4²)^{2}; 4^{4}]^{2} | 4-3,4-3,3-4/m90/r(h3) |

(3^{3}.4^{2})^{2}; 3^{2}.4.3.4 | 4-4,3,4-3,3,3-3,4-3-4/r/r(h17)/r(h18) |

3^{3}.4^{2}; (3^{2}.4.3.4)^{2} | 4-3,3-0,4,3/r/r(h2)/r(h18) |

[3^{6}; (3^{3}.4^{2})^{2}]^{1} | 4-3,0,3-3-3/r(h5)/r(h19)/m(h18) |

[3^{6}; (3^{3}.4^{2})^{2}]^{2} | 4-3,0,3-3/r(h3)/r(h15)/m(h14) |

[(3^{6})^{2}; 3^{3}.4^{2}]^{1} | 4-3-3-3-3-3/m90/r(h3) |

[(3^{6})^{2}; 3^{3}.4^{2}]^{2} | 4-3-3-3-3/m90/r(h2)/m(h22) |

3-UNIFORM (3 VERTEX TYPES) | |

3.4^{2}.6; 3.6.3.6; 4.6.12 | 12-6,4-3,3,4/m30/r(c5) |

3⁶; 3^{2}.4.12; 4.6.12 | 12-3,4,6-3/m60/m(c5) |

3^{2}.4.12; 3.4.6.4; 3.12^{2} | 6-4-3,12,3-3/m30/r(h2) |

3.4.3.12; 3.4.6.4; 3.12^{2} | 6-4-3,3-12-0,0,0,3/m30/r(c2) |

3^{3}.4^{2}; 3^{2}.4.12; 3.4.6.4 | 12-4,3-6,3-0,0,4/m30/r(h11) |

3⁶; 3^{3}.4^{2}; 3^{2}.4.12 | 12-3,4-3-3-3/m30/m(h9) |

3^{6}; 3^{2}.4.3.4; 3^{2}.4.12 | 12-3,4-3,3/m30/r(v1) |

3^{4}.6; 3^{3}.4^{2}; 3^{2}.4.3.4 | 6-3-3-4-3,3/m30/r(h10) |

3^{6}; 3^{2}.4.3.4; 3.4^{2}.6 | 3-4-3,4-6/m30/r(c5) |

3^{6}; 3^{3}.4^{2}; 3.4.6.4 | 6-4-3,4-3,3/m30/r(c5) |

3^{6}; 3^{2}.4.3.4; 3.4.6.4 | 6-4-3,3-4,3,3-3/r60/r(v5) |

3^{6}; 3^{3}.4^{2}; 3^{2}.4.3.4 | 3-4-3-3/m30/r(h6) |

3^{2}.4.12; 3.4.3.12; 3.12^{2} | 12-4-3,3/m90/r(h6) |

3.4.6.4; 3.4^{2}.6; 4^{4} | 6-4,3-3,0,4-6/m90/r(v5) |

3^{2}.4.3.4; 3.4.6.4; 3.4^{2}.6 | 6-4,3-3,3,4-0,0,6,3/m90/r(h17)/m(h1) |

3^{3}.4^{2}; 3^{2}.4.3.4; 4^{4} | 4-3-3-0,4/r90/r(h3) |

[3.4^{2}.6; 3.6.3.6; 4^{4}]^{1} | 4-4-3,4-6/m/r(c3)/r(h29) |

[3.4^{2}.6; 3.6.3.6; 4^{4}]^{2} | 4-4,4-3,4-6/m90/r(c5)/r(v1) |

[3.4^{2}.6; 3.6.3.6; 4^{4}]^{3} | 6-3,4-0,4,4-0,4/m90/r(h9) |

[3.4^{2}.6; 3.6.3.6; 4^{4}]^{4} | 6-4,3,3-4/m(h4)/r/r(v15) |

3^{3}.4^{2}; 3^{2}.6^{2}; 3.4^{2}.6 | 4-6-3,0,3,3-0,0,4/m90/r(h4) |

[3^{2}.6^{2}; 3.4^{2}.6; 3.6.3.6]^{1} | 4-6,4,3-0,3,3,0,6/m(h2)/m |

[3^{2}.6^{2}; 3.4^{2}.6; 3.6.3.6]^{2} | 4-6,4-0,3,3/r(h2)/m90/r(c9) |

3^{4}.6; 3^{3}.4^{2}; 3.4^{2}.6 | 4-6,4-0,3,3-0,3,3/r/r(c1)/r(h17) |

[3^{2}.6^{2}; 3.6.3.6; 6^{3}]^{1} | 6-6-3,3,3/r60/r(h2) |

[3^{2}.6^{2}; 3.6.3.6; 6^{3}]^{2} | 6-6,6,3-3,3/m//r(h8)/r(h49) |

3^{4}.6; 3^{2}.6^{2}; 6^{3} | 6-3-3/m/r(h3)/r(h15) |

3^{6}; 3^{2}.6²; 6^{3} | 3-6/r60/m(c2) |

[3^{6}; 3^{4}.6; 3^{2}.6^{2}]^{1} | 6-3-3,3-3,3-0,3/r(h7)/r(h29)/r(h29) |

[3^{6}; 3^{4}.6; 3^{2}.6^{2}]^{2} | 3-3,6-3/m/r(h6)/r(c6) |

[3^{6}; 3^{4}.6; 3^{2}.6^{2}]^{3} | 6-3-3/m90/r(h2) |

[3^{6}; 3^{4}.6; 3.6.3.6]^{1} | 3-3,3-3,6,3/m90/r(v1)/r(v15) |

[3^{6}; 3^{4}.6; 3.6.3.6]^{2} | 3-3-6-0,3/r60/m(c1) |

[3^{6}; 3^{4}.6; 3.6.3.6]^{3} | 3-3-6/r60/r(v4) |

[3^{6}; 3^{3}.4^{2}; 4^{4}]^{1} | 4-4-3-3/m90/r(h7)/r(v1) |

[3^{6}; 3^{3}.4²; 4^{4}]^{2} | 4-4-3-3-3/m90/r(h9)/r(h3) |

[3^{6}; 3^{3}.4^{2}; 4^{4}]^{3} | 4-4-3-3-3/m(h9)/r(h1)/r(v1) |

[3^{6}; 3^{3}.4^{2}; 4^{4}]^{4} | 4-4-3-3-3/m(h9)/r(h1)/r(h3) |

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**MDPI and ACS Style**

Gomez-Jauregui, V.; Hogg, H.; Manchado, C.; Otero, C. GomJau-Hogg’s Notation for Automatic Generation of k-Uniform Tessellations with ANTWERP v3.0. *Symmetry* **2021**, *13*, 2376.
https://doi.org/10.3390/sym13122376

**AMA Style**

Gomez-Jauregui V, Hogg H, Manchado C, Otero C. GomJau-Hogg’s Notation for Automatic Generation of k-Uniform Tessellations with ANTWERP v3.0. *Symmetry*. 2021; 13(12):2376.
https://doi.org/10.3390/sym13122376

**Chicago/Turabian Style**

Gomez-Jauregui, Valentin, Harrison Hogg, Cristina Manchado, and Cesar Otero. 2021. "GomJau-Hogg’s Notation for Automatic Generation of k-Uniform Tessellations with ANTWERP v3.0" *Symmetry* 13, no. 12: 2376.
https://doi.org/10.3390/sym13122376