Quadrilateral Lace

Year: 2020 Authors: Susan Happersett

Core claim

Simple mapping rules on squares, rhombi, and trisosceles trapezoids generate varied lace drawings with perceived spatial movement.

Topics

algorithmic drawing, point-to-point mapping, self-similarity, perception of space

Domains

geometry, quadrilaterals, coordinate patterns, mapping rules, generative art, lace drawing, ink on paper, pattern design

Methods

procedural mapping, layering concentric shapes, tiling, parameter variation

Media

ink on paper, quadrilateral grids, line segments

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2020 Conference Proceedings

Quadrilateral Lace

Susan Happersett

Beacon, NY, USA; fibonaccisusan@icloud.com

Abstract

This paper will describe the exact procedures I use to draw algorithmically generated mapping patterns in the framework of quadrilaterals. I began with a square format, then rhomboid and trapezoidal. Layering concentric self-similar shapes creates the illusion of 3-dimensional space. Shifting the self-similar shapes out of the concentric order creates the sense of movement across the plane.

The development of my lace drawings has been a multi-year process. By a lace pattern, or point-to-point mapping pattern, I mean carefully laying out a collection of points in the plane according to a pattern, and then joining certain pairs of points with line segments, where the pairs to be joined are specified by a definite procedure. The first lace drawings I algorithmically generated were restricted to points on the Cartesian Coordinate system: the $x$ -axis, the $y$ -axis and the lines $y=x$ and $y=-x$ [1]. Expanding on the types of point-to-point mappings I could produce, I decided to use the perimeters of quadrilaterals to assign my points. I began by working with a square with 7 equally distant points on each side. I picked an odd number of points for aesthetic reasons. Four sides on the square, 7 points per side, produces 24 points in total. I wanted a point on the center of each side. Once the parts were defined, I needed to write a rule to define the mapping process to be carried out for each point. I decided each point should be connected to the point 8 points (the number of points per side plus one) away in the clockwise direction. This mapping was completed for all 24 points as in Figure 1. The next quadrilateral I used for a structure was a rhombus - Figure 2 - with 120-degree obtuse angles and 60-degree acute angles. Again, I used 7 equally distance points on each side. The third quadrilateral I explored was a trapezoid with three congruent sides (I call it a “Trisosceles” trapezoid), Figure 3. The fourth side measures half the length of the other three sides. Again 7 points on each side, but on the shorter side the points are half the distance apart than on the longer sides.

Figure 1: $N=7$ Square

Figure 2: $N=7$ Rhombus

Figure 3: $N=7$ Trapezoid

To change the number of points on each side of the quadrilaterals from 7 to 5, 9, 11, … I needed to create some general rules; $N=$ the number of points on each side, $T=$ the total of points, then $T=4(N-1)$ . For the mappings, point $x$ maps to point $y$ but $y$ must be (less than or equal to) $T$ . In formulas:

(1) if $x\le 3N-5$ then $y=x+(N+1)$ (2) if x > 3N - 5 then $y=x+(N+1)-T$

Happersett

Using these formulas, I can draw lace patterns of different sizes, but of the same shape with the same distance between points. This layering technique creates some interesting perspectives in the drawings.

Figure 4: Spiraling Squares, Ink on Paper, 2020

“Spiraling Squares” (Figure 4) uses a grid of squares with 9 points on each side. Within each is an $N=7$ square and an $N=5$ square. The placement of these additional squares is alternatingly lining up with the perimeter points on the top left corner and the bottom right corner. This creates the sense of a spiraling motion.

“Cubic Lace One” (Figure 5) consists of an arrangement of three $N=9$ rhombi each with a $N=7$ and a $N=5$ rhombus positioned in its center. The altered perspective of this drawing offers a 2-D rendering of a cubes within a cube.

Figure 5: Cubic Lace One, Ink on Paper, 2019

Figure 6: Trapezoid See-Saw, Ink on paper, 2020

“Trapezoid See-Saw” (Figure 6) is a tiling of $N=9$ trisosceles trapezoids with $N=7$ trisosceles trapezoids situated so that the points on the smaller side of the $N=7$ trapezoids are centered on the points on the smaller side of the $N=9$ trapezoids. My exploration into point-to-point mapping patterns has allowed me to explore numerous possibilities in altering the perception of space and movement while still adhering to the restrictions of hand drawing on a flat piece of paper.

[1] S. Happersett. “Cartesian Lace Drawings.” Bridges Conference Proceedings, Waterloo, Canada, Jul. 27-31, 2017, pp. 391-394. http://archive.bridgesmathart.org/2017/bridges2017-391.