Adventures in Math Art

The other day I saw a design out of an old Arabian dictionary.  It was two interlocking squares, joined together by arcs into one continuous line.  It looked like something I could copy with a straight edge and compass, and in fairly quick order, I was able to work out the underlying geometry.

I worked on a color version in InkScape.

The next day, I wondered if one could do something similar with two interlocking pentagons.  Since it's slightly easier to make a pentagon in InkScape than with analog tools, I sat down at the computer and worked out how to place the arcs on the lines.  Since I was working with pentagons, I decided to use red as the main color. 

The next day after that, I wondered what other polygons would work with the interweaving straight lines and arcs arrangement.  I thought hexagons would be too tight for the arcs to fit aesthetically, and two interlocking triangles would result in "Happy Hanukah" jokes from Mark.  So I set out to work with a septagon.  

This is where I discovered that the figures with angles more acute than right angles don't allow arcs to nestle into their corners so easily.  I had to experiment with septagrams with rays of various thicknesses before finding one that would work.

The arcs, it turns out, will have an angle of 180 minus (360/number of corners in the figure) and be centered on the intersection where two lines meet.   I think the double-square pattern came out the best; the other two are fine... and I have a feeling I could nudge the radius of the arcs and the thickness of the lines some to make the "spokes" of the pattern more even.

I like them, and maybe I'll put them into a story. . . 

Small Stellated Dodecahedron

 

 

 

The latest adventure in not writing playing with graphics is a stellated dodecahedron.  Someone posted a picture of a marble floor mosaic from the Basilica of St Mark.  I'd seen it before, and I decided that I could probably copy it.  So I dug out my compass and ruler, constructed a pentagram, and went forward from there.

I switched to InkScape once I saw how the design is basically pentagonal rotations.

and added some etching lines.

 

I'm not sure which version works the best.  I like the etching lines--especially when the design is viewed from a distance, but I think the solid colors work better because the optical illusion of the triangle placement makes it look like I used more than six shades of grey.  

Latest Inkscape Art

Here's the latest design I've been working on.  If I were the sort of person that got tattooed, I might get this.  I might make it a T-shirt design. If I'm feeling extra creative, I might try flipping it so that the snake's head is at the bottom of the triangular knot, which might give the design a little more dynamism.

Designing Ankhs

The other day I was fiddling around with sketches, and I wondered if I made an ankh using a half-circle and a triangle made out of two 3 x 4 x 5 right triangles how it would look.  My sketch looked better than the design I came up with, so I must be bad at guestimaiting what kind of triangles I use.  Actually, I often find that the designs I come up with which appear to use circles, squares and triangles in my sketches look wrong when I use Inkscape to make geometrically accurate circles, squares and triangles.

Once I saw that the loop wasn't going to work with pure circles and triangles, I reworked it by tweaking the spline-points on a regular circle and then finding a triangle that fit the loop and used it as the serifs for the ankh's arms.

During the process of moving thick-lined polygons around, designing the arms to be hollow suggested itself, and I ran with it.

Virtual Stars and Labyrinths

One of the rewards I give myself is working with Blender to create 3D images.  Lately, I've managed to use InkScape to create 2D patterns which I then import into Blender extrude into 3D shapes.  

Importing from InkScape works fairly well with angular objects, but curved objects tend to generate objects with a broken manifold or with a missing facet that shows the insides of solid objects.   When an object isn't manifold, I have to go in and edit objects on a vertex and face levels.

Blender can use Boolean Logic to create complex shapes by combining or subtracting simple objects.  My favorite function is the difference function, followed by intersection function.  

One of the latest designs I've worked on are a Moroccan tile design created by interweaving five pointed stars.  Since the symmetry is pentagonal, the golden mean shows up frequently. Interweave designs require careful focus during the InkScape design process to get the over-under pattern correct.  

The other design is a labyrinth.  After looking at ancient and classical labyrinth mosaics, I worked on something similar.  After following some false trails (heh), I found that creating a roughed-out design on paper was more efficient than trying to make paths work in InkScape. 

I managed to create a labyrinth and get it into Blender.  Then I extruded it into something a person could walk through.  Or possibly a stove element.  

Revisiting the labyrinth, I managed -- with a whole lot of difficulty -- to grab just the top vertices of the extruded labyrinth, then scale them down so that instead of a disk shape, I had a cone shape.  From there, playing with a sphere gave me... Tron's lightcycle helmet, I think.

Oh well.  I'm hoping that I can create some interesting objects by sending the Blender Items to Shapeways, a 3D printing company.

Stars and Squares

 More playing with patterns.

I meant to use a zig-zag line instead of squares.  The end result looks more like a quilt design than zellij, but it's still pretty cool.  

What I like about tessellating and rotating squares like this is how the design starts to look like interlocked circles.

OK, and ordering squares around like this is tickling the part of my clever monkey brain that likes to manipulate things.

Rotating Zigzags Into Stars

The other day I came across a design that looked like it used only lines zig-zagging at right angles and I thought, "I could do that."

I made a mistake working from memory and forgot that I was supposed to be using a zig-zag and used squares instead.  But eventually I got the source pattern in front of me and saw my misstep.

Why yes, you could cut this design out on paper and then fold it into a truncated icosahedron; why do you ask?

More Holiday Craft

I managed to cutter-plot out some partridge card.  I think it looks better with an insert that make the pear look better.

I like the swan.  The designs work well with lights, although I wish it worked better than it appears to with dim tea lights.

Mark likes the goose.

I think in terms of lighting, these work out a little better with tissue paper behind it to help diffuse the light source.

What I learned the most about this process is that about two-and-a-half inches across is about as small as I can go with these designs before the design gets to small to produce on a plotter-cutter. (Pause to imagine using a laser-cutter...)

Um, yes -- the Twelve Days would fit on a dodecahedron.  Why do you ask?

Eye of Horus

I had some unexpected time so I went to the paper shop to use their paper cutter-plotter. I plugged in my Eye of Horus design.  I made a few eyes and a "St Teresa of Avila" style display frame. Unfortunately, after I got home, the large eye slipped behind a base-board heater and got a little bent.  I'm flattening it under a huge stack of books, so maybe I'll get better pictures tomorrow.



Mark insists this is a Free Mason symbol.  It's not exactly....  Well, OK, it does look like something from a Laura Croft movie.... Hmm, I was going to take this to work for my Shrine to Tea.




Oh well.  I can feel my Psychic Powers growing. That'll come in handy when I'm dealing with Graduates of the Eugene Psychic Friends Network School of Driving.