The Fascinating World of Tessellation: Exploring the Square Graph Paper

Tessellations have long fascinated mathematicians and artists alike with their intricate patterns and repeating shapes. One such example is the tiling of trapezoids and triangles, which bears a striking resemblance to the more common hexagonal and triangular tilings with six-fold rotational symmetry. Each pair of trapezoids fits neatly into one hexagon, resulting in six triangle twists around each pair.

This unique tessellation can also be observed in various real-world phenomena, such as the vertical orientation resembling a chain of buckets raising and lowering cargo or thrust faults and rolling waves of tsunamis. The slanted bars appear to be pushed into place by the rotating trapezoid twists, giving rise to an abstract machine-like pattern.

The alignment of patterns on graph paper is a crucial consideration when creating tessellations. While I have done the math to align this particular pattern, it remains untested in practice due to the lack of grid alignment. This challenge becomes more apparent with patterns like Woven Strips, where stripes do not align with the edges of the paper.

To address this issue, rotated grids can be employed to align the pattern repeats. However, this raises questions about what constitutes a 'square grid.' Is it aligned with the edges of the paper or is it an abstract concept applied to the physical space?

When folding tessellations, we are presented with various grid options, all of which align with the edge of the paper in some way. These choices highlight the invisible nature of our decision-making process when working with grids.

From a mathematical perspective, researchers have explored the properties of tessellations in various contexts. For instance, the facial Thue choice number of plane graphs has been studied using entropy compression methods (Przybylo et al., 2016). Additionally, scholars have examined the symmetry groups present in famous architectural structures like the Alhambra (Grunbau, 2006).

In conclusion, the world of tessellation is rich with fascinating patterns and mathematical concepts. By exploring the square graph paper, we can gain a deeper understanding of these intricate designs and their applications in various fields.

References:

Fecková Škrabuľáková, E., & Grešová, E. (2019). Costs Saving via Graph Colouring Approach. 45/2019.

Montesinis, J. M. (1987). Classical tessellations and threefolds. Accepted in Scientific Papers of the University of Pardubice, Series D.

Przybylo, J., Schreyer, J., & Škrabuľáková, E. (2016). On the facial Thue choice number of plane graphs via entropy compression method.

Tchoumathenko, K., & Zuyev, S. (2001). Aggregate and fractal tessellations.

Zabarina, K. (2018). Quantitative methods in economics: Tessellation as an alternative aggregation method.

Tessellation square graph paper