Labyrinths and space-filling curves

Labyrinths

A labyrinth is a single path that twists around and fills a shape.  Examples include the circular path at Chartres Cathedral in France and the rectangular El laberinto del patio by Ernst Kraft in Málaga, Spain.  The Labyrinth society has provided a website for locating a labyrinth close to you. Jo Edkins, known to many lacemakers for her online lace school, has written an excellent summary on the History of mazes and labyrinths.  Her website includes instructions for constructing several designs.

Labyrinths appear frequently in bobbin lace patterns.  Pat Brunsdon and Michele Minguin-Debray have designs based on the labyrinth at Chartres.  Christine Mierecki and Barbara Corbett both feature a square labyrinthine layout on the cover of their Milanese lace books.  The most prolific source is Hinojosa tape lace, such as this fabulous ring by Carolina de la Guardia.  Most Hinojosa patterns are comprised of a single tape that winds around with wonderful symmetry and sensitivity to form. The tape completely fills the shape, often with an elaborate outline.

Besides finding labyrinths visually beautiful, we associate spirituality and introspection with the act of tracing their path.  As you get turned around on the convoluted journey, you lose reference to the outside world. Your focus naturally turns inward to your own purpose and meaning.  This form of meditation is very familiar to lace makers.

Space-filling curves in Math

Mathematicians are also interested in these convoluted paths.   You may recall that a line has one dimension. A solid square is a two-dimensional object and a solid cube is a three-dimensional object.  Mathematicians in the 1800s wanted to know whether it is possible to draw a line that is so convoluted that it completely fills a square.  If so, does the line have one dimension or two, or perhaps something in-between?

In 1890, Giuseppe Péano, from the University of Turin, presented the following answer:

Divide a square into nine smaller squares.  Trace a single path that visits every one of the nine squares without crossing itself as shown in Figure 1a.  We will call this “path N”.  Number the nine squares in the order they are visited by path N (see Figure 1a).

Now subdivide each of the nine squares into nine smaller squares for a 9×9 grid of 81 small squares.  Our next goal is to place a copy of path N in each small square in such a way that the paths connect to form a single line.  Start by placing a copy of path N in the bottom-left square (see Figure 1b). Next place a copy of path N in the middle-left square.  Since we want to connect the two paths together with the shortest possible line, the second N must be flipped.  The first path N ends on the right. Once flipped, the second path N starts on the right and the points we need to connect align.

Continue adding N paths, flipping as required and following the order one to nine, to complete the pattern as shown in Figure 1c.  Repeat this process a third time to get the pattern shown in Figure 1d, on a grid of 729 very small squares.

peano

Figure 1: Creating a Péano curve: (a) First iteration, draw a basic path through 9 squares. (b) Second iteration, subdivide each square in Figure 1a into nine smaller squares. Draw a copy of path N in block one and a flipped copy in block two. (c) Repeat to complete the pattern. (d) Third iteration, subdivide each square in Figure 1c into nine smaller squares and repeat the process starting at the bottom left corner.

With each pass, the path becomes more convoluted and visits more points in the square.  If you repeat this subdivision process an infinite number of times, the journey will take you to every point in the square.  Therefore, given infinite time, you can draw a two-dimensional line!

For design purposes, the Péano curve does not have to be limited to a square space.  When curved into a circle, we end up with something similar to the labyrinth at Chartres (see work by Marie-Pascale Corcuff).

chartres

Figure 2: a) Labyrinth at Chartres, source: http://commons.wikimedia.org, b) First part of Péano curve bent around a circle.

Over the years, mathematicians have discovered additional examples of what they call “space-filling curves”, some of which are shown in Figure 3.  Each of these patterns can be made more and more detailed by a repetitive process.  In a future post, I will provide a tutorial on how to generate these fractal designs with Inkscape.

spacefilling

Labyrinths and space-filling curves in Torchon designs

Convoluted paths are also useful in Torchon for creating large pieces, such as shawls or scarves, using a small number of bobbins. For a rectangular Torchon piece, such as the example in Figure 4, the number of bobbins required can be reduced by half using the following technique:

  1. When designing the piece, choose a ground with 45º reflection symmetry to lie at each corner.
  2. Work the right half of the lace from the top down to the reflection line through the bottom right corner.
  3. Rotate your pillow 90º counter clockwise and continue working to the next reflection line.
  4. Rotate your pillow once more and work back to the top. Along the inside edge of this last section, sew the new work to the previous work.

 

torchon sewing pattern

Figure 4: An example Torchon pricking and pair diagram.  The pattern is rotated at red dashed lines. Line colours in pair diagram indicate which stitches are made before turning.  Red pinholes indicate sewings between yellow and blue sections.

If the ground uses symmetric stitches (stitches that look the same when turned 90º), the pattern will appear uniform, as if it has been worked in just one direction.  For very large pieces, you can use spirals, zigzags or a labyrinthine path to reduce the number of bobbins even further. For the pattern shown in Figure 5, I used the Hilbert space-filling curve (3rd iteration) to reduce the number of bobbins in the ground section from 64 pairs to eight pairs.

In her trilogy Discover,Explore,Master Torchon, Ulrike Voelcker describes this technique in more detail.  She provides several layouts [p. 123-20] including the top left corner of the “Wunderlich 1” pattern shown in Figure 3.  Voelcker also provides instructions [p. 117] on how to join two halves of a stitch with a nearly invisible sewing.

hilbert-curve_cropped

Figure 5: Hilbert space-filling curve as lace (high resolution image). Instructions on how to work this pattern will appear in Vuelta y Cruz/ Twist and Cross issue No. 29. On sale in August 2020.

If we only use one colour of thread, the labyrinth outline will disappear in the finished lace; only the lacemaker will know about it.  However, we can take advantage of the convoluted path and turn it into a design element by using two colours of thread: bobbins on the left side of the “tape” will hold colour A; bobbins on the right will hold colour B.  To keep colour A threads on the left,  use a turning stitch (C pin TTC) whenever a colour A pair and colour B pair meet.

The TesseLace ground used in Figure 5, was generated by my computer algorithm.  The pricking, working and thread diagrams in Figure 6 show how it is worked.  While a little more complicated than Rose ground, this TesseLace ground also has 45 degree symmetry.

thread-diagram

Figure 6: TesseLace ground (a) pricking, (b) working diagram (green intersections worked as Twist-Cross, rose intersections worked as Cross), and (c) thread diagram.

You can play with the crosses and twists used in this ground pattern by going to GroundForge – Diagrams of bobbin lace patterns.  Follow the instructions for modifying stitches.  I will write more about this useful ground exploration tool in a future post.

 

Fibonacci, quasiperiodic patterns and design

An abridged version of the following article appeared in the Bulletin OIDFA (Quarterly Journal of the International Bobbin and Needle Lace Organisation), 2020 No. 1.

When I design a pattern for bobbin lace, I turn to mathematics for creativity and inspiration.  In this article, I want to share a story of what this looks like.

One dimension

Imagine you are holding a lace edging that is several meters long.  When looking at the lace, you can find different spots that look the same.  Place your finger on one of these spots and ask a friend to place their finger on another.  Now compare the edging under your fingers as you both travel along the lace in the same direction and at the same speed.  You will see that the lace under your finger is always the same as the section under your friend’s finger until one of you reaches the end of the edging.  If the lace were infinitely long, you could keep doing this forever.  The lacemaker might have made a few mistakes or improved their technique over time so real lace is rarely exactly the same everywhere.  However, you can still see how they repeatedly worked the same pattern to create the yardage.

In math, a design that repeats by sliding the pattern in one direction is called a “frieze”, named after the frieze decorations found in architecture.  The motion of sliding is called a “translation” and the distance the pattern must slide to line up for the next repeat is called its “period”. Frieze patterns are periodic patterns in one dimension, so called because you slide the pattern one direction.

In Figure 1, you can see the pattern for a Torchon edging with diamond-shaped areas.  Some diamonds contain Rose ground (I have labelled them ‘A) and others have spiders (which I have labelled ‘b‘).  The diamonds repeat in the pattern AbA.

fig1_periodic_frieze

Figure 1: A traditional periodic Torchon lace edging

Does a pattern have to be periodic to repeat?  No!  Like the repetition of fallen leaves under a tree or birds soaring in a flock, you can create a design that repeats but not in the rigid way of a periodic pattern.  One way to do this is to use randomness.  For example, you could choose the order of the spiders and Rose ground by flipping a coin, heads means ‘make a spider’ and tails means ‘use Rose ground’.  If you make a long enough length of this edging, half of the diamonds will be A’s and half will be b’s but you could get runs of A’s and runs of b’s that are fairly long.  What if you would like to preserve the order AbA?  This is where a famous story from mathematics comes to mind: the problem of counting how fast rabbits multiply in ideal conditions.

In the 12th century, a mathematician, known as Fibonacci, imagined a reproduction rate for rabbits that went like this: you start with a pair of baby rabbits, one male and one female.  After one month the babies mature to adults.  By the second month, this adult pair gives birth to a new pair of babies, again one male and one female.  Their offspring also mature in one month and begin breeding in two months.  This cycle repeats month after month for all the descendants. (Aside: This ideal model leaves out many messy details such as death or two babies of the same sex. Simplifying a problem to make it solvable is something that is very common in math. )

To simplify things, I will use ‘b‘ to label a pair of baby rabbits and ‘A‘ to label a pair of adult rabbits.  The pattern of growth that Fibonacci described can be represented by a “substitution rule”: When a pair of baby rabbits reaches adulthood, we say “b becomes A”.  When two adults give birth to a new pair, we represent this as “A becomes Ab”.  This is what the first few generations would look like:

Day 0:                   b
1 month:              A
2 months:            Ab
3 months:            AbA
4 months:            AbAAb
5 months:            AbAAbAbA
6 months:            AbAAbAbAAbAAb
7 months:            AbAAbAbAAbAAbAbAAbAbA
and so on …

If we count how many pairs of rabbits we have each month, we get the famous Fibonacci sequence of numbers: 1 1 2 3 5 8 13….  Each number (except for the first two) is the sum of the previous two numbers (e.g., 13 = 5 + 8).

OK, that is interesting but what does it have to do with designing lace?

Mathematicians call a sequence of symbols a “word”.  For example, the sequence of A’s and b’s that we used to describe the rabbit population is a “Fibonacci word”.

Look at the Fibonacci word for the 7th month.  You can see that the pattern AbA repeats many times.  Sometimes, like at the beginning of the word, two copies are side by side (AbA AbA) and sometimes, like at the end, two copies overlap (AbAbA).  Mathematicians have proven that a Fibonacci word never has two b’s in a row or three A’s in a row.  Also, the pattern will never repeat periodically.

We can use the Fibonacci word to choose what to put in each diamond of our lace edging, as shown in Figure 2.  Now if you and your friend trace this edging, starting from two spots that look the same, at some point there will be a difference between what each of you is looking at.  Mathematicians call this kind of repetition “quasiperiodic” or “almost periodic” which means that a pattern (AbA in our example) appears repeatedly, but the larger piece cannot be created by sliding a pattern along with a regular period.

fig2_quasiperiodic_frieze

Figure 2: The beginning of a quasiperiodic Torchon lace edging that uses the Fibonacci word

You can also use the Fibonacci word to place tallies on Torchon ground.  In Figure 3, I assigned each symbol from the Fibonacci word to a crossing of pairs.  Going from left to right: if the symbol was A, I left the original crossing; if the symbol was b, I placed a red square to represent a tally.  When I arrived at the end of a row, I continued from the left on the next row.  A similar approach can be used to arrange snowflakes in a Binche design.

fig3_tallies

Figure 3: An arrangement of tallies based on the Fibonacci word

As a Fibonacci word gets longer, the ratio of A’s to b’s gets closer to the Golden Ratio.  This magical ratio is often associated with beautiful proportions.   The Fibonacci word is only one of many that belong to a family called the Sturmian words.  This family also includes the Tribonacci (three different letters) and Thue-Morse words.   Any of these words can be used to design lace.

Two Dimensions

But the story doesn’t end there.  Let’s take a look at bobbin lace grounds, which are also patterns that repeat many times in our lace designs.  To fill a shape in our design, we slide the ground pattern in two directions.  In mathematics, a pattern that repeats in two directions is called a “wallpaper” pattern.  We can also find wallpaper patterns in printed fabrics, tiled floors, and brick walls.

Now the interesting question is: Can we design a bobbin lace ground that is not periodic but still contains many copies of the same pattern? The answer is: Yes, we can!  Two more families, from the many two-dimensional quasiperiodic patterns known to mathematicians, will now join our story.

First, let me introduce the Ammann bar family.  These patterns are like the half-stitch ground (also known as triaxial weave).  In half-stitch ground, the cross-twist stitch weaves threads in three directions.  Each set of parallel threads is at a 120 degree angle to the other sets.  In the 1970s, a mathematician named Robert Ammann noticed that you could arrange five sets of parallel lines in an interesting pattern, shown in Figure 4.  For each set of parallel lines, Ammann used the Fibonacci word to determine the distance between lines.  He used ‘A’ for a long distance and ‘b’ for a short distance.

fig4_ammann_bars

Figure 4: Ammann bar pattern with 5 sets of lines.  Two of the sets are highlighted, one in green and one in black.

In the lace shown in Figure 5, I worked one line from each set with pink thread.  Where the five lines come together, they form a pentagram (which is inside a pentagon inside another pentagram inside …).  Other patterns in the Ammann bar family have four or six sets of parallel lines.

small_fig5_ammann_bar_lace

Figure 5: Ammann’s web: lace made from the pattern in Figure 4 by Veronika Irvine

The second family, called the quasiperiodic rhomb tilings, contains the popular Penrose tiling.  A rhomb is a shape with four sides that are all the same length.  Like the Fibonacci words, a substitution rule is used to fit the rhombic tiles together in a quasiperiodic pattern. Figure 6 shows the Ammann-Beenker tiling.

fig4_ammann_beenker

Figure 6: Ammann-Beenker tiling in grey with lace pattern marked in red

In this figure, look at the corner of a rhomb and count how many line segments meet at that corner.  In many places, this gives an odd number. As lacemakers, we would prefer to have 4 line segments to meet at every corner because this corresponds to two pairs of threads coming together to make a stitch then going apart. So we can’t make lace directly from these beautiful patterns.  However, there is an easy way to convert a pattern made of rhombs into a lace pattern.  To start, place a dot at the center of every rhomb.  For each rhomb, draw lines from the dot at its center to the dot at the center of every rhomb that it shares a side with (see the red lines in Figure 6). Because a rhomb has four sides, there will always be four line segments connecting this dot to the other dots.  In Figure 7, you can see the resulting lace pattern.  Notice that it contains many copies of the same pattern (see blue highlights).  I have worked this pattern using a grand Venetian cord (yellow threads) for some of the stitches to emphasize the repeating elements.  You can explore many more rhomb tilings in the Tiling Encyclopedia.  With three pair crossings and four pair crossings, you can also turn patterns with hexagons or octagons into lace using the same trick.

fig5_ammann_beenker_lace_pattern

Figure 7:  Lace pattern derived from the Ammann-Beenker tiling

small_fig8_ammann_beenker_lace

Figure 8: Nodding bur-marigold: lace worked from the pattern in Figure 7 by Veronika Irvine

As you can see, math provides valuable tools for creative design. For more information on the concepts that I have presented here, please read my mathematical article on quasiperiodic bobbin lace grounds.  If you would like to see more of the patterns I have created using these and other mathematical principles, please visit my tools section.  You will find free patterns to download and use in your own creations.  I would love to hear about any lace you have made using math.

References

“The Fibonacci sequence: A brief introduction” by  Rachel Thomas
https://plus.maths.org/content/fibonacci-sequence-brief-introduction

“The golden ratio and aesthetics” by Mario Livio
https://plus.maths.org/content/golden-ratio-and-aesthetics

“The Mysterious Mr. Ammann” by Marjorie Senechal
http://www.squaring.net/downloads/ammann-senechal2004.pdf

“From quasicrystals to Kleenex” by Alison Boyle
https://plus.maths.org/content/quasicrystals-kleenex

Tilings Encyclopedia
https://tilings.math.uni-bielefeld.de/substitution/

“Quasiperiodic bobbin lace patterns” by Veronika Irvine, Therese Biedl, and Craig S. Kaplan
https://tesselace.com/research/quasiperiodic-bobbin-lace/