EscherMath Fall 2009

A shiny near-spherical dream catcher…

 A pentagonal hexacontrahedron is, as the name implies, a sixty-sided solid with pentagon faces.  The pentagons are irregular with three short sides where three faces meet at a vertex and two longer sides meeting at a more acute angle where five faces meet.  I created this platonic solid out of two layers of cereal box glued together, cut out, hot-glued, and painted.  For the exterior design, I used my knowledge of tessellations and personal experience in creating knots to draw the pattern of one side, which I then scanned.  I used Adobe Photoshop to duplicate the outline within the template for a 5-side part of the hexacontrahedron net, which I found on a website provided in class.  I printed out the 12 templates and glued them onto the double-thick cereal box – an easily and cheaply scrounged oft-recycled material that I chose because it is light, durable, and easily manipulated.  The next day, when the Elmer’s was dry I started to cut 

out the negative space of the knot with scissors and exacto knife.  

This was a fun, time-consuming process, which made me very glad that I had not chosen a more dense material.  After a few days when I had cut out one-half of the units, I hot-glued them together to make sure it would be stable – especially because my design eliminated the join of one of every three of the short sides – where pairs of green loops almost touch.  As I was successful in constructing the first hemisphere, I finished cutting out the remaining six parts and hot glued those together as well.  While it was still physically possible, I painted the interior a uniform blue so all of the cereal graphics would not give away my choice of material.  After gluing the two hemispheres together – a task that was, in fact, made much simpler by the holes in the design, I started to paint the exterior.  This process, detailed though it seems, took a fraction of the time of cutting out the units.

             I discovered while painting that my design, which I planned with two contrasting strands, actually consisted of many links instead of the intended two overlapping knots.  I’m used to planning knots in 2-d and did not fully take into account the differing properties of 3-d tessellation and symmetry.  When my project was fully complete, I braided embroidery floss to make a harness so I could hang my shiny near-spherical dream catcher above my bed.  It’s rather hypnotizing if you twirl it like this…

Time Warp by Robin Kresge

This piece is meant to represent the fourth dimension. Most believe the fourth dimension is time/space, so I drew an elderly woman with a mirror of an identical, younger version of the woman. Where their hands are touching, their hands are warped into each other, with a warping line extending to a cuckoo clock. The face of the clock is also warped. By this, I meant to represent the subject’s movement through time, almost through the “warp line.”  In drawing this piece, I used 4H drawing pencils, a kneaded eraser, and a sketch pad.

        For my project I made a hyperbolic tessellation with an ideal 6-gon.  It is ideal because the sides of the 6-gon hit the edge of the circle.  I started with one ideal 6-gon and added more as I worked my way outside until it was too small for me too continue.  I divided the ideal 6-gon into six triangles. Hyperbolic geometry used geodesics which are lines that hit the edge at 90 degree angles.  So all of my lines in this tessellation hit the side of the circle at 90 degree angles.  The schafli sign for my tessellation is {3,6}.  Meaning that each shape has three sides, and six of the shapes are connected at the vertices.  The dual of my tessellation has a schafli symbol of {6,3}.  After I drew the tessellation I colored it.  I used a pattern of dark to light colors with blue and green.  For my materials I used thick artists papers, pencil, and colored pencils.  Before drawing my final piece I drew many practice ones to make sure it looked okay.  I was inspired to do a hyperbolic tessellation because I really enjoyed learning about them, and drawing them in class.  So, I thought it would be fun to create a more complicated one than we did in class. 

Knot theory is the mathematical study of knots which began in the early 19^th century.  It has become more enthusiastically studied in the past ten years as its application to physics and fundamental forces that created the universe around us was discovered.  For example knots are involved in the enzymes interaction with DNA.  

Initially I wanted to make a knot that formed a five, pentagonal, or six, hexagonal, pointed shape.  I put trefoil knots between each point. There only exist two types of trefoil knots and each one is unique.  They are left or right trefoils, determined by which side of the strand overlaps first, the one on the left or the one on the right.  If looked at in isolation my trefoil knots are “left” trefoils.  I liked the way trefoils looked so I wanted to make a complex knot that incorporated them in it.  I took out the original pentagon shape I had sketched in the center of my knot, because it didn’t connect correctly to the rest of the knot.  Then I plotted my sketch out on graph paper to make it symmetrical and the elements even.  Also, this allowed me to determine the x, y, z coordinates of each crossing.  I transferred the graph paper image of my knot to artist drawing paper; then used colored pencils to draw it and indicate the crossing points by breaking the line of the strand that was behind the other strand.  Next, as suggested in much of my reading on the subject, I created a model out of a piece of string which represented a twisted braid made out of 3 threads each. These formed the braids of the knot.  Then I sewed each crossing and twist, where the strands touched each other, and painted the knot with white acrylic paint, so it would maintain its shape for the presentation.  This helps one to visualize the knot in its 3 dimensional form.  Note that the five flower-like branches off the center of the knot are not knots, because they can be untwisted to create a single loop.  The trefoil shaped branches off the center cannot be untwisted but form part of the complex knot which includes the center of the piece.  Because my knot can be represented in 3 dimensions it is called a tame knot, those that are not are called wild and exist in dimensions beyond the third dimension.  This is one of the ways knots are classified; others are the number of crossings it has, whether it’s a mirror image of itself and the type of braid within the knot.  My knot has 35 crossings, it is not a mirror image of itself and it has 3 twisted threads that make up each braid.  I indicated the math for each crossing on the graph diagram of my knot.  I assumed that the z coordinate for each point was 0 and 1.

My artwork is a fractal like design, laid over a simple tessellation of squares which just acts as a background of air. The fractal is based around a central perfect hexagon.  Then on each side of the hexagon there is another hexagon attached to the side, half of the side of the original. Then on each of these smaller hexagons there is another hexagon, half of the new hexagons size, on the three sides farthest away from the central hexagon.  Then on the outer hexagons (two per each secondary hexagon) there is one half sized hexagon on the farthest side of the hexagon, then one more half sized hexagon on the farthest side of the quaternary hexagon. Then on the central tertiary hexagons there are three half sized hexagons. Lastly each of these quaternary hexagons has three half sized hexagons on the three sides farthest from the corresponding secondary hexagon. The larger central hexagon acts as a trunk. The secondary and tertiary hexagons act as branches. Finally the quaternary and fifth-level hexagons act as leaves; all of the hexagons then add together to form a tree. The art work is about the relationship between dimensions. The idea that a tree can be viewed from the perspective of a two-dimensional object, gives a little bit of an idea of how something in four dimensions might look to us simple three dimensional beings. The inspiration for this artwork came from a combination of two things: fractal trees and the book Flatland. Fractal trees are very simple fractals that are really cool; they are simply lines that after a certain distance branch off to form other lines going on into infinity and none of the lines ever touch.  Flatland is a book from the perspective of a two dimensional square, the book concerns his run in with a three dimensional object, a sphere. The sphere appears as a small circle out of nowhere and as it passes through the plane, grows bigger, then shrinks, and eventually disappears.  This idea of how a three dimensional object appears to a two dimensional object, makes it much easier for three dimensional objects, such as humans to conceptualize the fourth dimension, and how four dimensional objects would appear in the third dimension. The mathematic qualities involved in this art are: fractals, the dimensions, and perspective.  Fractals are patterns that demonstrate self-similarity and set repeated alterations of size; this is seen in the hexagons getting smaller and smaller but still being proportional. The dimensions and perspective are seen through the fact that the tree is a three dimensional object experienced from the perspective of a two dimensional object. For research the book Flatland was researched.

For my final project in Escher Math class, I decided to research a topic that I found very interesting during the course of the year, fractals. They always made me think critically and were an excellent representation of how something mathematical can be represented physically in a beautiful and captivating way. I spent too much time fiddling with the Mandelbrot viewer after my homework was done and later found images of other fractals, like the Phoenix and Julia sets. Fractals are images or objects that are self-similar. Self similarity exists in an object if the object “may be written as a union of rescaled copies of itself, with the rescaling isotropic or uniform in all directions”. That is, if we zoom in on the object, we begin to see the same image all over again, and again, and again. This introduces a concept called scale invariance, which means that there is no natural size to any fractal, and as such, in a true self-similar fractal, we can never see the image as a whole because no matter how big or small our view is, we still see the same image. Self-similarity is evident in natural landscapes, mathematical fractal sets (like the Mandelbrot and Julia sets) and in naturalistic fractals as well (fractals designed to replicate or look like nature, for example, fractal trees or artificially created ferns). Of course, perfectly self-similar fractals are very rare (a good example is the Sierpinski Triangle) so we use another term to describe fractals that are not perfectly self-similar or at least, contain a weaker form of self-similarity: Self-affine. For instance, many researchers find fractal patterns present in clouds and landmasses, but if we were to zoom in very close to these objects, the resultant image may not even be remotely recognizable when compared with the original; these are self-affine. Fractals that are self-affine, or display a random pattern of duplication, are called random fractals. Some mathematicians will also refer to fractals as “exactly” self-similar, “quasi” self-similar”, or “statistically” self-similar depending on their degree of self-similarity (from strongest to weakest).

“The Mane Knot”

For my project, I decided that I was wanted to incorporate the knot theory into my artwork.  I live on a farm and I enjoy drawing horses so I decided to use that as my subject.  I decided to turn the horse’s mane into a knot.  I was using a variety of colors and I decided this horse would work much better as a unicorn.  It took some time to figure out how I was going to work the knot into the mane.  I also turned the border into two knots.  For my materials I used colored pencils and poster board.