Doodle No. 37 Three Cats of Dr. Caligari

Never be afraid to go full rainbow. 

Prismacolor black ink, Faber-Castell Polychromos pencils, Finetec and other mica watercolor paint on Stonehenge 250 GSM 100% cotton paper

4”square

See the flash of the mica.

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Doodle No. 22 Bunny and Owl

The image is 3” by 3”. 

The paper is 4.5" by 4.5". 

Drawn with archival Prismacolor black ink and Faber-Castell Polychromos pencils on Stonehenge 250 GSM 100% cotton paper. 

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Highly Unlikely Dodecahedron in Bugle Beads and Crystal

I received a commission to make a highly unlikely dodecahedron, and after what seemed like a billion stitches, here it is.

 

A quick off-the-cuff estimate suggest that this piece contains around 3000 beads.  It's roughly the size of a baseball. Ten different colors of beads show the ten paths that twist and wrap around the surface.  Each path encircles one of the ten axes of 3-fold rotation symmetry of the dodecahedron. (Those axes go through opposite vertices of the dodecahedron so that the paths are like belts that twist around the equators of those axes.)

If you want to learn more about how I discovered this shape, check out my blog post on Highly Unlikely Triangles and Other Impossible Figures in Bead Weaving.  There, you can find a link to download the paper I presented at the Bridges Conference in 2015.

I don't have a tutorial for the dodecahedron, but I do have a tutorial for a related and simpler design, the Highly Unlikely Triangle, shown below with bugle beads, but you can also use just seed beads. Thanks for looking.

 

Bridges Paper - Highly Unlikely Triangles and Other Impossible Figures in Bead Weaving

I have been going a little crazy saving a surprise for you all, and today is finally the day to share it!  Meet the Highly Unlikely Tetrahedron.

These are photos that I will be presenting with my paper at the Bridges Conference in Baltimore this month.

My paper is called, "Highly Unlikely Triangles and Other Impossible Figures in Bead Weaving." It is now available from the Bridges website. You can download the free PDF file here.  I hope you enjoy it!

Be sure to browse the entire collection of papers in the 2015 Proceedings of the Bridges Conference.  There are so many great contributors this year; it will be impossible to pick a favorite. Seriously, go brew yourself a pot of coffee, start at the beginning and click on any title that looks interesting. Candy for your brain. If you can make it to University of Baltimore on July 29 - August 1, 2015 (Wednesday - Saturday), you can even go to the Bridges Conference and listen to the authors talk about their papers, which is way more fun than reading them because you can meet the people who wrote the papers and ask them questions.

If you don't know about the Bridges Conference, neither did I until about 2003, when I heard a talk at the Joint Mathematics Meetings by Reza Sarhangi, a Bridges founder and generally fascinating fellow. Reza talked about this meeting where people discuss connections between mathematics, science and the arts, including visual art, architecture, music, poetry and theater.  As I listened, I thought, "I think I have found my people." It was a big moment for me, listening to Reza speak.  So, I went home and wrote a couple short papers on quilting and math, and submitted them for review.  Next thing I know, I was presenting my work to a group of like-minded people, other mathematical artists and mathematicians who loved art. They had many fascinating ideas to share, and they educated each other, plus they had interest in my work and opinions about it. I was in math-art-nerd heaven. I went to four Bridges conferences in so many years, and then I left academia to be an artist, and stopped going, and started going to Burning Man instead and doing art there with that community. Then, after last year with the Genie Bottle, I decided to take a year off of Burning Man, and go back to Bridges this year instead. So I wrote a paper on beading impossible figures, they accepted it, and I booked my tickets.

Then, Kelly Delp and the other organizers sent me an email. They thought my paper was so swell that they asked me to give a keynote address to the whole conference.  That means that I get more time to talk and show slides, and there will be no other concurrent sessions while I will be speaking.  I also get a little spot on their website here among the other keynote speakers, including John H. Conway, Ingrid Daubechies, and Alan C. Kay, who all have their own Wikipedia pages, by the way. So, you get that I'm excited to go and see and meet all the people. They asked me to make a mosaic for their website, which you can see here. 

The mosaic includes photos of our jungle gym Bat Country and the Genie Bottle, two Burning Man art projects that I created with the help of my friends in Struggletent.  The beadwork includes a circular Celtic knot, my cover of the Journal of Mathematics and the Arts with an Octahedral Cluster and a Seirpinski tetrahedron, bacteriophages, DNA, a Highly Unlikely tetrahedron from my Bridges paper, and a few photos of cellular automata, which is the project I'm currently working on.  So stay tuned for that. It's going to be cool. You'll like it.  I promise.

I hope to see you in Baltimore!  As always, thanks for looking.

Highly Unlikely Triangle and Squares

The Highly Unlikely Square and Triangle are based upon the optical illusion known as the impossible triangle of Roger Penrose, made popular by M.C. Escher in the 1950s. The Swedish artist Oscar Reutersvärd drew the first impossible triangle in 1934.

These little beaded objects look nice strung on a piece of cord or chain to be work as a small pendant.

I have a tutorial for the Highly Unlikely Triangle, if you would like to learn to make one yourself. 

Or, if you want to just have one and not make one, all of these pieces are for sale in my Etsy shop.  Click on the photos to go to the listings.  As always, thank for looking!

New Tutorial - Highly Unlikely Triangle with seed beads and thread

I finally finished a tutorial for the Highly Unlikely Triangle, now available.

This piece of beadwork is based upon the impossible triangle of Roger Penrose and MC Escher. I beaded my first impossible triangle in 2006, and it’s taken me until 2014 to write this tutorial. This wasn't my first try, or second!   Until recently, I couldn't find a good way to explain how to do it on paper.   In those 8 years, I have been developing my tutorial writing skills to the point that I could manage explaining such a tricky and challenging design in a way that I think advanced beginning beaders will understand.  And now, I think I finally got it.  The solution was lots and lots of illustrations.

This tutorial is designed to teach you cubic right angle weave with seed beads, including turning corners and joining ends to make a closed loop.  You will also learn to embellish CRAW with seed bead, on straight-aways, and also around inside and outside corners. You can even use tiny bugle beads, like I show here.

 Or even longer bugle beads, like I did here.

I hope you will enjoy my newest tutorial of one of my oldest designs.  Thanks for looking.

Math Anxiety Camp: An emotional art piece

For the last few years, I have been attending a festival in the middle of the Nevada desert called the Burning Man Art Festival, where tens of thousands of artists and revelers join together each summer to celebrate art while camping in some of the harshest conditions in the US.  One of the peculiar features of Burning Man is that groups of people band together into "theme camps" which offer free gifts in the form of art, music, food, drinks, experiences, and other life pleasures and fancies to almost anybody who stops by.  Many of the gifts come with the touch of a prankster, designed to shock and amuse. Last year, we created the idea of Math Anxiety Camp, where we would give people math problems in an attempt to provoke math anxiety.  Burning Man has a culture of gift giving, and we joked, "Math anxiety: It's our gift."

because math matters

Great art provokes emotions.  This is a reason why music is so popular and powerful.  Music provokes intense emotions in listeners.  Have you ever cried to a love song? I have.  Now, I have never seen a piece of artwork that was designed specifically to provoke math anxiety.  So I created Math Anxiety Camp with the help of my camp mates.  For Burning Man 2013, we wrote a short book of 38 math problems, and I designed a sticker so we could hand out awards to those who achieved math anxiety.

Math Anxiety Camp Achievement Award

My campmates and I drilled unsuspecting Burning Man attendees (i.e., burners) with our math problems in the hopes of provoking the emotion of math anxiety for the sake of art.  We did all of the things you're not supposed to do as a good math teacher, like telling our examinees, "You should already know this," "You should have learned this last year," and "Work faster! Faster, faster, FASTER!"  When participants got wrong answers, we made loud buzzer noises.  The purpose was not to focus on the math, but to focus on experiencing and emphasizing the emotion of math anxiety.

Those who are tardy don't get a fruit cup (Thank you Kimberly Laabs)

Vi Hart and I wrote the book of the math problems together with several features in mind.

Vi and me sitting on Bat Country

All of our math problems were designed to be actual math problems that have at least one right answer (some have more).  Topics included arithmetic, combinatorics, geometry, calculus, and logic. Problems ranged in difficulty from trivially easy (e.g., "Name a number that is 3.") to tricky (e.g., "Name a triangle with two right angles.") but all were chosen to be simple enough that most of our subjects would at least understand the question, even if they couldn't solve the problem.   Some problems were designed to be funny.  We included several classic, well studied math problems that are known to confuse people.  Most of the problems have multiple choice answers, and the distracters (incorrect wrong answers) were designed to be funny or deliberately confusing or deceiving.  We included lots of "All of the above" and "None of the above" options because of their cringe value. We added scenarios relevant to the art festival, and where the characters were in mortal danger.

My campmates eagerly distributed math problems, books, and awards throughout the festival.  I was pleasantly surprised at how many people engaged in the project.  We handed out nearly 500 achievement award stickers and almost 20 math books to specific people who wanted to own a copy.  I listened to and heard about several people who read the entire book, thoroughly musing over each the 38 problems.  Some were math teachers, married to math teachers, physicists, geologists, engineers, and others who just enjoy the satisfaction of solving a good math problem.  Here is a PDF copy of the 2013 Math Anxiety Math Book if you'd like to take a look.  This version is edited to make it more suitable for a general audience.

Ethan Port brought a whole suitcase of math books, and Paul McGlaughlin painted us a sign for the front of our camp so that passers by would know we were there and what we had to offer.  As one young man biked past, we overheard, "Math Anxiety Camp!  That's TERRIFYING!"  So of course, we invited him in to share our problems and win an achievement award.

Camp Sign
Photo by Daniel Thornton

When people asked us about our sign, they often confused us with "Math Camp at BRC", a different theme camp that touts themselves as "a safe place for mathematics." We were the opposite: an unsafe place for mathematics.  We were there to give problems, not solve them. "Our problems are your problems," and, "We have so many problems, we'll give you some!"  I saw people tense up instantly when they heard these statements.  Although we gave them lots of problems to solve, and many people successfully solved them, my favorite part always was watching their anxiety transition into laughter when they were presented with anxiety achievement awards.

Paul painted half of the sign with chalkboard paint so we could write a new problem each day. We left chalk by the board, and many of the problems were solved by the next morning.

Question of the Day
Photo by Daniel Thornton

In addition to working independently, Math Anxiety Camp also joined forces with Camp UFOm and the Civil Defense Camp.  UFOm provided an "interblastive foam experience" where participants performed foam art.  UFOm was so popular, that the Civil Defense crew was enlisted to conduct drills on the revelers in an attempt to slow down the line of entry into UFOm and deter all but the most dedicated from entering. 

Civil Defense Bunker, Tent and Ropes Course

Burners were subjected to drills including physical exams such as a rope course, running laps, push-ups, jumping jacks, rolling in the dust, and games of duck-duck-goose.  There were also oral exams on outdated American civil defense literature from the 1940s, and people were drilled with math problems from the Math Anxiety Camp Math Book.

Civil Defense Drills: Photo by Ben Harper

A little background on the Math Anxiety Camp project:  I am a former teacher of mathematics, and the idea of provoking math anxiety on purpose is simply ridiculous to me.  I spent many years of my life trying just about anything to minimize, or at least reduce math anxiety in my students because students who are too anxious don't perform well in school.  Math anxiety makes people hate math and avoid it.  Wanting people to love math and engage, I read many papers on math anxiety, attended lectures on the subject, wrote worksheets and led discussions in my classrooms that were designed to reduce my students' anxiety.  It seemed to be an ever-present problem in my classrooms of college students, many of whom had learned to fear math from a very young age, typically spawned from negative interaction with their teachers and parents.  I was often surprised at how quickly some of my students were to state their disdain for mathematics publicly, even though they were studying for professions that would require them to do mathematics regularly, like engineering or teaching children.

My experiences as a math teacher showed me just how common math anxiety really is and how intensely some people suffer from it.  Some people will go to great lengths to avoid math at all costs just to avoid the anxiety that goes with it, and this makes me sad.  But outside of the context of teaching, it seems that math anxiety is an emotion that is rarely discussed in depth, especially in the art world.  Creating math anxiety in a novel context devoid of high-stakes consequences seemed like a good way, a safe way for people to confront their negative emotions about math.

Math has been a theme for me and my camp mates already at Burning Man.  This year, Math Anxiety Camp was also an art support theme camp, building Bat Country, a Sierpinski tetrahedron jungle gym.  Here you can see Bat Country this year on the night of the Man burn.

Bat Country
Photo by Daniel Thornton

 Here is Bat Country with the Rainbow Bridge art car.

Bat Country and the Rainbow Bridge

Significantly, we were not able to elicit math anxiety in all of our subjects.  Many participants easily and eagerly solved our math problems without anxiety.  It's not terribly surprising that burners were quick to engage in the idea of Math Anxiety Camp.  My sense is that burners are more mathematically literate than the average American population, which probably correlates with the maker attitude of the festival participants.  In addition to Math Camp at BRC, there is a thriving tradition of beautiful mathematical art at Burning Man.  My favorite returning piece this year is the honorarium art project Zonotopia and The Quasicrystalline Conjunction by Rob Bell

Zonotopia and The Quasicrystalline Conjunction

The mathematics behind these "pavilions" is polar rhombizonahedra, one of my all time favorite mathematical structures.  I've been watching this series of inhabitable structures evolve, changing from year to year, but still maintaining its same aesthetic and mathematical essense.  This portion below was the new addition to the set for 2013.  The panels have a lot more details than most of the older forms made in earlier years. Beautiful. 

My favorite new piece of mathematical art this year is the honorarium art project, The Penrose Triangle by Blake Courter and Blake Courtney. 

The Penrose Triangle

 This triangle looks very different from different perspectives.

The Penrose Triangle

Unfortunately, I missed a shot of the triangle in perfect perspective where all of the lines look straight, but this one is pretty close.

The Penrose Triangle

Fortunately, I climbed up to the top of this triangle and successfully climbed back down without killing myself. Burning Man always has a plethora of climbable objects, and I love to watch the acrobats and other "monkeys" climb and hang off these piece.  However, I rarely climb anything at Burning Man, which is a little ironic since I brought my own jungle gym (see Bat Country above).  Although I don't generally suffer from math anxiety, I do suffer from a fear of heights (or high anxiety), but I don't really want to talk about that emotion.

Pavel Curtis sent me this link.

Highly Unlikely Squares and Blueberry Earrings

Red, it's one of those colors that I generally don't like much.  For once, I was truly inspired by red, and with turquoise no less! 

I was inspired to use this color combination from the bracelet shown below.  This bracelet was designed and beaded by the very talented Mikki Ferugiaro (photo used with permission).  Thank you, Mikki, for the great color scheme and the inspiration! You actually made me like red, and that's no small accomplishment.

Here's another close up of the earrings.

Do they look huge? Last time I saw Florence in person, I was wearing my own Mini Blueberry Beaded Bead earrings (in purple and green), and she was very surprised by how small and dainty they are.  She had seen photos on line, and she always imagined them to be much larger than they really are.  Mini Blueberry Beaded Beads measure just 11 mm in diameter.  In my book, that definitely qualifies them as minis.  See how small...

While I had these seed beads out, I made a second Highly Unlikey Square pendant that matches the first, but with green instead of red.  What can I say?  My love for red is fickle, and by now, you know how I feel about green.  At the same time, I actually like all colors (except yellow) because I've learned that any color can be beautiful if it's used well with other colors, even red.  Now, if I can just figure out how to love yellow, I'll be set.

I have a tutorial for the Highly Unlikely Triangle, a related but simpler design. 

Thanks for looking.

Highly Unlikely Hexagon and Borromean Link

I'm still having fun with cubic right angle weave.

This piece is Three Intersecting Rectangles for the Math Nerd In You No. 4 (Borromean Link).  Each of the three rectangles is an entirely separate piece of bead weaving.  Although all three are linked together, no two are linked to each other.  If you would like to learn how to make one for yourself, check out the pattern and kits for the beaded Borromean Links. 

Here is a Highly Unlikely Hexagon, also made with cubic right angle weave.  Matte black, gold and aqua... and I added a few sparkling Swarovski crystals at three of the corners for a little subtle bling.  These pieces are available in my Etsy shop, just in case you want them.     

 

I have a tutorial for the Highly Unlikely Triangle, a related but simpler design.  Thanks for looking.

Unlikely Square Earrings and Pendant

I've been fiddling around with cubic right angle weave (learn how to make CRAW), making beaded versions of things like the Impossible Triangle of Roger Penrose.  This strange construction was made famous by M.C. Escher, and several years ago, I made what I think was the very first one using bead weaving.  Since then, several new finishes of beads have been made available, many of which have found there way into my big box of beads.  So I decided to revisit these designs with some of my newer bead finishes.  I'm really quite fond of these new colors.

Because a square has four sides, and CRAW also has four sides, you can color the square frame with four different colors that wind their ways around the frame.  Each color does one lap around the square.  Here's a different color scheme of the same construction.

The triangle doesn't lend itself to this same kind of coloring, since it has just one edge and one face that wind their way around the triangle three full laps. I have a tutorial for the Highly Unlikely Triangle. 

I really wanted to use more colors than in the triangle above, so I made this triangle below.  It has a different color on each leg of the triangle.

I really liked the color scheme of the square earrings and pendant at the top of this post, so while the beads were out, I made some DNA Earrings (learn how to make them yourself).  I added some green amethysts to add a touch more sparkle.

And a little pair of Cutie Pie Earrings (learn how to make them yourself), with three natural sparkling amethyst on each earring.

Links for the tutorials are in the text above.  The jewelry is for sale in my Etsy shop.   Click on the photos to see the listings.  Thanks for looking.