Math Embroidery: Tessellations

Tessellations are designs created by lining up geometric shapes edge to edge, with no over lapping or gaps.  They are commonly used in tile designs and fabric motifs.  Soccer balls use tessellation to cover a sphere with 2D straight edge polygons.

Six strand cotton on 20 ct linen.  Back stitch.

 Tessellation is used often in quilting design.  Below is a close up of a quilted piece I did using the same motif detailed in the photos both above and below.  I used back stitch, instead of running stitch.

Quilting cotton on cotton fabric.  Back stitch.

Six strand cotton on 20 ct linen.  Back stitch.

Tessellation can be used to solve mathematical problems visually.  For example, Voronoi Tessellation uses data points to create a diagram that shows geographic distance.  The areas shaded represent all the points that are the equidistant from each other.  City planners use Voronoi Tessellation diagrams to determine how far apart shops can be and still impact each other.

Six strand cotton on 20 ct. linen.  Back stitch.

Penrose Tiling is unlike traditional tessellation in that it doesn't have a repeating pattern.  Instead, it relies on reflected symmetry to create a self-similar structure.

Six strand cotton on 20 ct. linen.  Couching, back stitch, chain stitch and split stitch.

Six strand cotton on 20 ct. linen.  Back stitch and chain stitch.

Lydia's Blanket

Detail of embroidery on blanket; six strand cotton on fleece

Detail of crochet edging on blanket

Lydia's blanket

I took a break from the math embroidery for a few days to finish Lydia's blanket in time for her second birthday.  The colors aren't super crisp in the photos; it is a light yellow fleece blanket.

Math Embroidery: Platonic Solids

Six strand linen and cotton
floss on 20 ct linen.
Same for all other photos.

Anyone who has played with dice or looked at pictures of the Egyption pyramids has seen the five Platonic solids.  They are found in art and decoration from every known culture.

They are called 'Platonic' because Plato wrote about them and identified each solid with one of the elements.  The element, earth, was associated with the hexahedron (cube) because Plato considered earth stable, for example.

My embroidery is based on an old textbook illustration.  I choose it because I liked the unfolded, 2D shapes below each 3D representation.

Doing this piece got me thinking about how most of our representations are 2D; artists and drafters have developed techniques for showing 3D on a 2D surface.  In elementary school, most of us are taught to drawn a path or river by diminishing the lines towards the horizon.  Those that wear glasses probably have noticed that their 3D vision is weaker as they look to the far edges of their lenses.  We call video games 3D even though they are depicted on a 2D screen on a TV or computer monitor.
We move between the 2D and 3D world pretty seamlessly in our every day lives. We experience what Relativity teaches is the 4D, time, in only one direction.

 If you think of time as an axis on a Cartesian graph, we only move in the positive direction.  Our memories and our language give us ways to express the negative axis of time, but we don't physically move along the negative axis on our Newtonian earth.

If String Theory can explain Quantum and Relativity Theory in one unified theory, then perhaps there are other dimensions that we aren't aware of.  Will artists of the future learn to depict more dimensions by using techniques like horizon -- techniques we can't begin to image?  Or will math formula and computer generated geometric abstractions be the only way we have to express these other dimensions around us?

Math Embroidery: Archimedes Find Pi

Three circles w/ polygons
Six strand linen and cotton floss on 20 ct. linen

Close up of purple Archimedes transcribed circle
six strand linen and cotton floss on 20 ct. linen

Archimedes, an ancient Greek mathematician, developed a method for finding Pi (necessary if you want to find the area of a circle or cone) that can be used with n-sided polygons. These polygons are inscribed (inside the circle) and circumscribed (outside the circle).  N just represents the biggest number you want to use; Archimedes used a 96-sided polygon and the formula for the area of a right triangle to came up with a value for pi as approximately 22/7.

I didn't attempt a 96 sided polygon because I am sure that on 20 count linen it would look like 3 circles.  Instead, I used 5, 6 and 8-sided shapes.  There is a bit of black work in each polygon.  The yellow/orange figure uses darning stitches but in black work design.  Typically, darning is done with thread the same color as the background fabric because you wouldn't want to see the area of patched fabric.  But darning stitches can be used in black work designs in contrasting colors.

Math Embroidery: Venn Diagrams and Set Theory

Four Ellipse Venn Diagram
6 strand cotton and linen floss on 20 count linen

Five Ellipse Venn Diagram
6 strand cotton and linen floss on 20 count linen

I am working on a set of embroidery pieces that detail math concepts using geometry.  The first set I completed are Venn diagrams as related to set theory.

I am no math scholar, but I am combining research, embroidery and lectures via podcast to try to teach myself some of the large concepts in mathematics.  

I understood how Venn diagrams worked before beginning the piece; high school teachers use Venn diagrams often as graphic organizers.  But set theory was new to me.  I learned that entire sets of data can be treated like individual values and manipulated.  For example, a simple proof uses set theory to show that the set of real numbers are infinite.  Take each even number (2, 4, 6...) and match it with each counting number (1, 2, 3...).  You could make a 1:1 match with group and find that you have the same amount of counting numbers and even numbers.  Now you might ask, what about the odd numbers?  If you think about the -set- of even numbers as counting numbers minus odd numbers, which logic tells us must be less than all counting numbers.  But because we can match the set of even numbers to the set of counting numbers, this shows that numbers, both counting and even (and odd) are infinite.  Mathematicians do this with sets of all types of numbers to show they are infinite.  Then using the fact that infinity exists, they go on to explain more complex things.

All of my patterns are based on wiki commons images.

Next up, Archimedes finds pi and the five platonic solids.

Voodoo Charms

Lust
six strand cotton on 22 count linen

I just finished a series of voodoo charms, or veves.  They represent the symbolism that voodoo practitioners use to call loa, or spirits of the dead, during complex rituals.

Voodoo is a mixture of west African tribal religions and Catholicism.  Just like in any active religion, there is no consensus on exactly what symbols and colors mean.  I took some artistic privilege and adapted the patterns from a series of charms that can be found here.

In voodoo, colors have symbolic meaning.

Red -- war, masculine energy and habits
White -- wisdom
                                                     Green -- abundance, nature
                                                     Blue -- harmony, water, psychic energy
                                                     Black -- death
                                                     Pink -- children, feminine energy, fortune

Protection

Snakes represent potential energy and arrows are used to direct energy.  Priests and priestesses will draw arrows in sand, corn meal, grave yard dirt or other dusty materials to guide spirits.

In the veve to the right, there is a cross in the middle.  It is believed that black, or jet, crosses ward off the evil eye by metaphorically stabbing the eye.

Winning in Battle

This veve represents Ogun, a loa associated with battle.  He is symbolized with sabers and is notorious for his over-the-top masculine behavior.  Ogun must be appeased with run and cigars and will smack pretty girls on the butt with his sabers.  (There's no hidden meaning there!)









Catholic Saints typically are described as having perfected personalities and are reserved in emotion. Voodoo loa, often identified with Catholic saints, amplify human emotions and desires.  They have big personalities and seem, to me, to make the point that one has to take the good with the bad.  For example, Brigitte, associated with finding lost things and lost children, is greedy and selfish.

Good Fortune

Lust close-up

Reap Power

Reap Power close up

Voodoo and Persian Tulips

I've been practicing stumpwork (3D embroidery) techniques for a week.  I have been working on using wire and button hole stitch to create tulip petals that are raised on the embroidery work.  However, I still need practice.  In the meantime, I've posted a tulip from my stumpwork sampler.  It comes from Jane Nicholas' book using Persian, Syrian and Indian artwork for inspiration.  

Below is part of a separate series I am working on inspired by voodoo charms and symbolism.

Purse Experimentation

Crochet, double stitch and double post stitch.  Acrylic yarn.  Brand unknown.

 I am still working on purse construction.  This attempt was made using a basic pouch design, but I added sides and a bottom to the pattern in attempt to give the bag more structure.  I wanted it to keep it's shape when carried.

The purse is reversible.  In the pictures above and to the right I am showing the 'wrong' side of the work.

This is the 'right' side of the work.  I didn't weave in the edges, because I am unsure if I want to keep the fringe as a design element or not.

Baby Staley Blanket

Crochet edging

My brother and wife are expecting a baby boy in September, so I made him a fleece blanket.  All his ducks are in a row...get it.  Ducks?  In a row?

Anyway, made just like the blankets for my Goddaughters, I embroidered the fleece; the doubled the fabric and secured with a blanket stitch.  The blanket stitching is also the foundation for the crochet edging.

Six strand cotton floss on fleece using back stitch

Two Embroidery Projects Finished

Six strand cotton floss on 20 ct linen

Notebook Cover

Mandala

Six strand floss on 14 ct. aida

Sculpy bead

More Purse Experimentation

Crocheted purse using Flounce yarn.  Colorway:  grays and white

I like how this purse turned out better, but I am still experimenting.  For the next purse I want to work on lining it.

Middle Kingdom Napkins

Below is a detail for the Middle Kingdom napkins I worked on for SCA.

Six strand cotton and silk floss on linen

Ugly Folded Purses

I've been working on purses for the past week, based on the Masa Bag .

First, I found the ugliest yarn I could find and crocheted this purse.

The yarn, being cheap and acrylic, gives the purse structure.  But it is difficult to see the folded pattern that it is obvious in the Masa Bag.

I found a bit of blanket that I started and never finished and used it for attempt 2.

I like this one a bit better because you can see the folding in the design.  But the softer yarn  makes the bag loose it's shape when worn.  I'll probably keep working at the design and embellishments but my next project must be my SCA napkins for Pennsic.

Olivia's Canvas Bag

And the last gift is finished!  Olivia's bag was embroidered on canvas using six strand floss.  I will also fill it with goodies.  The cake has been ordered and I'm ready for some birthday fun!  Have a great weekend!

Six strand floss on canvas.

Six strand floss on canvas.  Back stitch, Bayuex stitch and satin stitch (not shown).

Gabrielle's Tote Bag

I'm almost finished with the Guglielmo girls' birthday presents.
Below is a tote bag I embroidered for Gabrielle.  I am putting a couple of beginner cross stitch kits inside, along with other goodies.

Tote bag.  Six strand cotton floss on canvas.

Detail of flower on tote bag.  Six strand cotton floss on canvas.  Back stitch,  Bayeux stitch,  satin stitch (not shown).

Alexandra's Blanket

I finished Alexandra's blanket yesterday.  The colors a little difficult to see in these photos, but the blanket is a light green fleece.

Fleece blanket (light green).  Blanket stitch and crochet edging using cotton yarn.  Embroidered initial using both perle cotton and six stranded floss.

  Detail on Alexandra's blanket.  Six strand and perle cotton.

Edging on blanket.  Cotton yarn.

Sophia's Blanket

Detail of initial on fleece baby blanket.  Six stranded cotton floss.

My friend Anne had twin daughters a little over a year ago.  In honor of their first birthday, I am making fleece blankets with each girls' initial embroidered in the lower left hand corner.  The fleece has been edged in crochet.  Below is Sophia's blanket.

Fleece blanket.  Six strand cotton floss for initial.  Acrylic and cotton yarns for crocheting edging.

Close up of crochet edging.  Acrylic and cotton yarns.