With winter upon us, it's time for for some of my favorite symmetrical art: paper snowflakes!

Anthony Herrera has a site with all kinds of awesome Star Wars snowflakes. You can download pdf files of the designs and cut them out yourself. (This give me all kinds of ideas for other designs...)

My 9-year old daughter and I just tried two of his designs. Here's our Yodas and X-wings.

## 21.11.12

## 20.11.12

### Math of Vegetables on Schrödingers Katt

I was on TV last week, talking about the mathematics of vegetables
on a popular science show called Schrödingers Katt. It airs on NRK, the
national broadcasting service. We filmed in September and they had a pretty quick turnaround on production.

Here's the link if you'd like to see, it links directly to my segment. They also show me juggling throughout the closing credits. Juggling a pineapple, cabbage and cauliflower was no easy thing!

http://tv.nrk.no/serie/schrodingers-katt1/dmpv73003112/08-11-2012#t=21m3s

Here's the link if you'd like to see, it links directly to my segment. They also show me juggling throughout the closing credits. Juggling a pineapple, cabbage and cauliflower was no easy thing!

http://tv.nrk.no/serie/

## 2.11.12

### Factor Visualization

Dots are added one at a time and they all reconfigure to make patterns which display all of the prime factors of the new number. 70 for example, is shown as 7 groups of dots in a regular heptagon arrangement. Each of the groups is 5 groups in a regular pentagon arrangement. And each of these 5 groups is a set of 3 dots in an equilateral triangle. 105 = 7 x 5 x 3. The 7, 5 and 3 are clearly visible. The animation is mezmorizing.

I did a series of screenshots and made this composite image of the integers 1-49.

## 13.10.12

### 7-sided dice. With activities!

I spotted these dice in a hobby shop and couldn't resist. Seven sides!

These bring many questions to mind. First and foremost: are they fair? How can it be determined?

Trying to calculate by shape, angles or area would be insanely complicated. This could be resolved by experiment, though, by throwing them a large number of times...

Here's a thought experiment. These are pentagonal prisms, two pentagon faces with connecting rectangles. If the rectangles had very small thickness, the dice would be like coins and only land on one of the two pentagon faces. If the rectangles were very long, they'd be like pencils, and only roll one of the five rectangle faces. There must be a point somewhere in between where the probability of a rectangle or a pentagon are the same! Did the makers of these dice find that point?

Here's another thought: does the chance depend on the area of faces? Do equal areas = equal probabilities?

While we're thinking of prisms with equal area faces... well that's interesting in itself. What would a square prism with equal area faces look like. A cube. What about a triangular prism? Say the triangle has edge length 1, what edge length would the rectangle need to have equal area? (The answer is interesting!)

Notice that these dice have dot patterns split across the edges because an edge will be up. Clever design! The dots are colored to help distinguish them from the other "faces". Clever again! The pentagons have digits printed on them instead of dots patterns. Oh. Too bad. Was this a design choice to emphasize the "seven-ness" of the dice? I think dot patterns would look nice.

These bring many questions to mind. First and foremost: are they fair? How can it be determined?

Trying to calculate by shape, angles or area would be insanely complicated. This could be resolved by experiment, though, by throwing them a large number of times...

**Activity 1**: determine if the dice are fair.Here's a thought experiment. These are pentagonal prisms, two pentagon faces with connecting rectangles. If the rectangles had very small thickness, the dice would be like coins and only land on one of the two pentagon faces. If the rectangles were very long, they'd be like pencils, and only roll one of the five rectangle faces. There must be a point somewhere in between where the probability of a rectangle or a pentagon are the same! Did the makers of these dice find that point?

Here's another thought: does the chance depend on the area of faces? Do equal areas = equal probabilities?

**Activity 2**: devise a thought experiment/extreme case to disprove this.While we're thinking of prisms with equal area faces... well that's interesting in itself. What would a square prism with equal area faces look like. A cube. What about a triangular prism? Say the triangle has edge length 1, what edge length would the rectangle need to have equal area? (The answer is interesting!)

**Activity 3**: Investigate other equal area face prisms... any interesting patterns?Notice that these dice have dot patterns split across the edges because an edge will be up. Clever design! The dots are colored to help distinguish them from the other "faces". Clever again! The pentagons have digits printed on them instead of dots patterns. Oh. Too bad. Was this a design choice to emphasize the "seven-ness" of the dice? I think dot patterns would look nice.

**Activity 4**: Design dot patterns for all of the faces. (What dot patterns look good on pentagon faces? If not 6 and 7, then redesign all of the dot patterns so that each roll looks good).## 3.10.12

### Research days - magnetic building sticks

At the end of September every year Norway celebrate scientific research with a slew of national events. Here in Trondheim we have two days of a 'Research Fair' where folks from education and industry set up stands inside two huge tents in the middle of downtown. Friday we welcome schools and Saturday we're open for the public. This year's theme was "Society". The Math Center (my workplace) teams up with the Math Institute every year, and this year our team met several times to create building materials so kids could make polyhedra and learn about how polyhedra contribute to art, arcitecture and science.

We glued magnetic balls to the ends of 50 cm long sticks, and they worked remarkably well as building materials. We decorated our stand with posters showing polyhedra in art, architecture, science and everyday living, and we also constructed a newspaper geodesic dome that we placed on top of our stand (see earlier post for instructions). We had three tables and a carpeted area for building. For two days we built stuff with kids and it was fantastic fun.

We found three different designs for towers. Height was limited by the strength of the magnets, but we were able to build two of the designs to heights over 2 meters. Amongst the Archimedian solids we built: tetrahedra, octahedra, icosahedra, and a truncated tetrahedra (I thought it would be impossible, but we did it), as well as several cube variants though they required different structural supports because of the unstable nature of squares.

Here's some pics of the construction process. We bought flowers sticks from a craft shop and Bucky Balls, strong magnetic balls that are awesome to play with by themselves. With a glue gun, put a small dab on the end of a stick, press a ball into it, and after it has cooled place a small amount of glue on the seam and then rotate the stick while smearing the glue around the seam with the tip of the glue gun. The glue should reach up the the midpoint of the sphere or a little higher. If done properly (it takes practice!) there is only a small amount of glue but the magnet is fastened strongly to the stick.

To be determined by further research: is it better to have the poles parallel or perpendicular to the stick? My feeling is that perpendicular will make for a better building experience. The first picture below shows the magnet being attached parallel to the stick. This gives less opportunities for sticking to other magnets, though the attraction will be stronger when it does align. I am unlikely to build another set so I may not know, but should you wish to make a set, you might want to try both and let us know!

We glued magnetic balls to the ends of 50 cm long sticks, and they worked remarkably well as building materials. We decorated our stand with posters showing polyhedra in art, architecture, science and everyday living, and we also constructed a newspaper geodesic dome that we placed on top of our stand (see earlier post for instructions). We had three tables and a carpeted area for building. For two days we built stuff with kids and it was fantastic fun.

We found three different designs for towers. Height was limited by the strength of the magnets, but we were able to build two of the designs to heights over 2 meters. Amongst the Archimedian solids we built: tetrahedra, octahedra, icosahedra, and a truncated tetrahedra (I thought it would be impossible, but we did it), as well as several cube variants though they required different structural supports because of the unstable nature of squares.

Here's some pics of the construction process. We bought flowers sticks from a craft shop and Bucky Balls, strong magnetic balls that are awesome to play with by themselves. With a glue gun, put a small dab on the end of a stick, press a ball into it, and after it has cooled place a small amount of glue on the seam and then rotate the stick while smearing the glue around the seam with the tip of the glue gun. The glue should reach up the the midpoint of the sphere or a little higher. If done properly (it takes practice!) there is only a small amount of glue but the magnet is fastened strongly to the stick.

To be determined by further research: is it better to have the poles parallel or perpendicular to the stick? My feeling is that perpendicular will make for a better building experience. The first picture below shows the magnet being attached parallel to the stick. This gives less opportunities for sticking to other magnets, though the attraction will be stronger when it does align. I am unlikely to build another set so I may not know, but should you wish to make a set, you might want to try both and let us know!

## 18.9.12

### Bridges Gallery 2012

## 17.8.12

### Math Munch

I met a lot of great people at the Bridges conference this year, among them Justin Lanier, Paul Salomon and Anna Weltman, authors of the Math Munch blog. They did a short write up on my stuff that you can find here: http://mathmunch.wordpress.com/2012/08/06/mike-naylor-math-magic-and-mazes/

Be sure to check out the other stuff they have on their site. Good stuff!

Be sure to check out the other stuff they have on their site. Good stuff!

## 11.8.12

### The Human Kaleidoscope

Here's my film that debuted at the Bridges International Math Art Conference in Baltimore, MD, July 2012.

## 27.7.12

### Ballooning with Vi Hart

Vi Hart ran a mathematical balloon sculpting workshop at the Bridges conference in Pécs Hungary summer 2010. Here's the effervescent Vi with an icosahedral balloon sculpture, and me inside her group project Sierpinski pyramid. Squeaky fun!

## 24.7.12

### Pi acknowledgement

In Sandra Kring's novel "Thank You for All Things" (amazon.com link), one of the characters is an autistic child who memorizes pi and is close to the world record. Here's an excerpt about one of his practice sessions:

Find out more about sound numerals here: http://folk.ntnu.no/krill/home.htm

As I pass Milo’s door—eight hours after he began—he’s resetting his timer. He sees me and calls out, “I’m just starting position 48,551,” he says. “I’m averaging 6000 digits per hour. Right up there with the current record holders,” He sets the timer down and starts, “three,The part he likes, 3 7 2 5 4, is my name in sound numerals. A cool 'secret reference'.seven, two,five, four—I really like that part!—eight, two,five….” The sounds of Grandpa Sam’s rutted breaths and rattling bed bars are filling every corner of the house. Milo hears them too, of course, and I can tell by his eyes that he’s grappling hard to see the digits, rather than to see Grandpa struggling to breathe. I feel sorry for him, so I say, “Good job. Catch you later.”

Find out more about sound numerals here: http://folk.ntnu.no/krill/home.htm

## 20.7.12

### Human Hypercube

Speaking of hypercubes, here's representation with people. We can build up to a hypercube by starting with a 0-dimensional point. 'Stretching' that in one direction gives us a 1-dimensional line segment. Stretch the line segment in a direction at 90° from the line segment and we get a 2-dimensional square. Stretching that 90° from the plane of the square gives us a 3-dimensional cubes. Now, stretch the cube at 90° from its volume (yes, I know it hurts to try to think like that. It's only impossible in the real-world, not in your mind!) and you'll have a 4-dimensional hypercube.

The pictures above show the changes from 0-d to 4-d.

Puzzles:

1. Describe how the number of vertices changes. Write a formula.

2. Describe how the number of line segments changes from one figure to the next. Write a formula.

3. Describe how the number of squares changes. Yep, formula.

4. Describe how the number of cubes changes.

5. Predict the properties of a 5-d hypercube. Can you draw one?

## 17.7.12

### Nidaros Cathedral Hypercube

This lamp (one of several) in front of the Nidaros Cathedral in Trondheim bears a striking resemblance to a hypercube.

## 13.7.12

### Magic Static Cat

Two pieces of transparency film printed with static, taped on opposite sides of a piece of glass. From the right angle, the static aligns and the Cheshire Cat appears. This is hanging outside of my office in the elevator lobby. It's fun to point out to visitors. How does it work?

The trick lies in starting with 2 identical copies of static and inverting the static in one of the images in the area you want to appear black. That way, when the copies are aligned the regular static is still just static, 50% gray, but in the other area the black pixels are aligned with the clear pixels and vice versa, so that part is dark.

Note that inverted static holds as much information as non-inverted static, that is, zero. Neither of the sheets of static contain an image of the cat. It's only by juxtaposing the two images that the information can be generated/retrieved.

The trick lies in starting with 2 identical copies of static and inverting the static in one of the images in the area you want to appear black. That way, when the copies are aligned the regular static is still just static, 50% gray, but in the other area the black pixels are aligned with the clear pixels and vice versa, so that part is dark.

Note that inverted static holds as much information as non-inverted static, that is, zero. Neither of the sheets of static contain an image of the cat. It's only by juxtaposing the two images that the information can be generated/retrieved.

## 10.7.12

### Skolelaboratoriet Logo

Skolelaboratoriet is a department that is housed on the same floor as Matematikksenteret where I work. My first day on the job I was pleased to see their logo:

Can you see what is special about it?

It show a dissection of a square into an equilateral triangle. The white piece in the middle stays in place and is common to both the square and equilateral triangle. The three upper pieces which are part of the square are moved below, each with a 180° rotation, to create the equilateral triangle on the bottom. I've never tried to work out the angles and lengths involved to accomplish such a dissection, but it looks like a nice little problem. I'll try to remember to think about this next time I have a long flight.

The dissection is even a little better than a simple cut-and-rearrange: the pieces can be hinged so that one form turns inside out to become the other.

George Hart presents a coffee table that performs this trick:

## 6.7.12

### Cobblestone scallops

These scallop patterns in the cobblestone in downtown Trondheim are interesting. If you were laying cobblestone, how would you determine the layout for these stones? Using your technique, what other interesting curvy designs are possible?

## 3.7.12

### Lady in Glass

Peter Sutton is a glass artist in Trondheim, Norway. He and I collaborated on this project, a lady in glass:

I designed 23 cross-section slices which he printed on sheets of glass and assembled into an elegant block. The result is a 3d model floating in a solid chunk of glass. Here's a view of the cross-sections:

## 1.7.12

### Extra second, and celebrating a gigasecond

Last night an extra second was added to world time. It happened at 23:59:60 UTC, or Universal Time which is the same as Greenwich Mean Time minus daylight savings. Here in Norway we're at GMT +1 which is UMT +2, so it happened just before 2:00 a.m. here.

I stayed up late to celebrate. It was an eerie feeling, to experience a minute go by that was 61 seconds long. It dragged on and on and on and we began to wonder when this minute would ever end. Whoa. Good thing they only happen every 4 years or so.

Reminds me of when I turned one billion seconds old. It happens sometime when you're 31 years and some months old. I remember thinking about it when I turned 31 and I calculated the date. It was 252 days away and I soon forgot about it.

One morning I got an email from a guy I hadn't from in 10 years. The email simply said "Happy Gigasecond!" I panicked! I'd forgotten! Was it today? Had I missed it? I furiously calculated. I was born in Indiana at 1:02 p.m., now I was in Washington, I had to account for time zone changes and daylight savings. I found out the exact second... and I still had 4 hours to go. I celebrated by popping open a malty beverage at the exact second while in my hot tub. It felt good to be aware of and celebrate the exact second.

But I still had a mystery – how did this guy know? I wrote him back to thank him. Turns out 10 or 15 years ago at a juggling club meeting we all figured out the exact dates we'd turn one billion seconds old. Dave L. had written down all of these dates in his calendar, and being a very organized person at the end of each year he would buy a new calendar and transfer all of the relevant information from the old calendar to the new. This he did again and again, and when it came to be my turn, he tracked down my email address.

Dave, you're my hero! If I'd missed it I would have needed to wait until 63 to celebrate the next one.

I stayed up late to celebrate. It was an eerie feeling, to experience a minute go by that was 61 seconds long. It dragged on and on and on and we began to wonder when this minute would ever end. Whoa. Good thing they only happen every 4 years or so.

Reminds me of when I turned one billion seconds old. It happens sometime when you're 31 years and some months old. I remember thinking about it when I turned 31 and I calculated the date. It was 252 days away and I soon forgot about it.

One morning I got an email from a guy I hadn't from in 10 years. The email simply said "Happy Gigasecond!" I panicked! I'd forgotten! Was it today? Had I missed it? I furiously calculated. I was born in Indiana at 1:02 p.m., now I was in Washington, I had to account for time zone changes and daylight savings. I found out the exact second... and I still had 4 hours to go. I celebrated by popping open a malty beverage at the exact second while in my hot tub. It felt good to be aware of and celebrate the exact second.

But I still had a mystery – how did this guy know? I wrote him back to thank him. Turns out 10 or 15 years ago at a juggling club meeting we all figured out the exact dates we'd turn one billion seconds old. Dave L. had written down all of these dates in his calendar, and being a very organized person at the end of each year he would buy a new calendar and transfer all of the relevant information from the old calendar to the new. This he did again and again, and when it came to be my turn, he tracked down my email address.

Dave, you're my hero! If I'd missed it I would have needed to wait until 63 to celebrate the next one.

## 26.6.12

### Burning calories

There was a discussion at lunch today about how eating a big chocolate bar can require 2 hours on an exercise bike to burn it off. I've always found these comparisons a bit misleading. It makes it sound like that's the only way to keep from getting fat.

Fact is, a 100 g of milk chocolate has about 520 calories, which will indeed take 1 to 1.5 hours to burn off on an exercise bike. Or you could just sleep for 7 or 8 hours and it would also be burned off.

So go ahead and have that candy bar! Just be ready to either cycle for 2 hours or sleep for 8 hours. OR... sleep for 10 hours and lose weight. Mmmm... chocolate + naps = weight loss. I think I could start a very popular weight loss program.

Fact is, a 100 g of milk chocolate has about 520 calories, which will indeed take 1 to 1.5 hours to burn off on an exercise bike. Or you could just sleep for 7 or 8 hours and it would also be burned off.

So go ahead and have that candy bar! Just be ready to either cycle for 2 hours or sleep for 8 hours. OR... sleep for 10 hours and lose weight. Mmmm... chocolate + naps = weight loss. I think I could start a very popular weight loss program.

## 22.6.12

### Shadow knitting pi

Here's a 3 meter long shadow-knitted pi scarf made by Anne Bruvold in Tromsø. Shadow knitting creates patterns that are only visible from an angle. Seen straight on, as in the middle picture, the pattern is invisible.

Lovely! Anne used 3 plus 14 extra digits, which is my favorite length of
pi: 3.14159265358979. Try saying those digits a couple of times, they
nicely roll off your tongue like a poem.

Black on white digits viewed from an angle |

The digits are invisible viewed straight on |

A portion of the back of the scarf |

## 19.6.12

### Natural rhombic dodecahedra

My kids and I like to go up near Trolla and find garnet crystals in the rocks near the water or cliffs. The stone cracks open easily with hammer blows, and dark red garnet crystals spill out. The crystals are in the shape of rhombic dodcedahedra – polyhedra with 12 faces which are rhombuses. The shape is very special, it is the shape you'd get by putting a pyramid on each of the faces of a cube so that the faces of the pyramids are in the same plane as the neighboring pyramids. The resulting rhombuses have diagonals in the proportion 1:√2. A bunch of these crystals the same size would pack together tightly to fill space, which is pretty cool also. Not many shapes will do that. There's lots of other properties of this shape and connections to cubes and octahedra.

Here's some pics during and after one of our hunts:

Here's some pics during and after one of our hunts:

## 15.6.12

### Erno Rubik

At the Bridges conference in Pécs, Hungary in 2010, we had a rare presentation from Erno Rubik, inventor of the Rubik's cube. Earlier that day I'd gone shopping and tracked down a couple of cubes and a permanent marker, and hit him up for a couple of autographs afterwards. It was a high point of geek culture!

## 12.6.12

### Almost a Reuleaux, real Reuleaux and 3D Reuleaux

I saw this Nokia trinket on a colleague's cellphone. It's close to a Reuleaux triangle, but not quite. Darn! A missed opportunit from the folks at Nokia.

A Reuleaux triangle is a shape with constant width. If you roll it, you can set a board on top of it and the board will not bob up and down, it will stay at the same height just as if the roller were circular. You can make a triangle like this easily, just draw three circles that pass through the centers of each other, and the region in the middle is a Reuleaux triangle as shown here:

You can make these with 5, 7, 9 or any odd number of segments by constructing arcs on regular polygons.

George Hart made these solids of uniform width. "Reuleaux acorns" he calls them. They're a 3d generalization of Reuleaux triangles - if you place a flat board on top of them they will roll around smoothly and the board won't wobble up and down. Weird...

A Reuleaux triangle is a shape with constant width. If you roll it, you can set a board on top of it and the board will not bob up and down, it will stay at the same height just as if the roller were circular. You can make a triangle like this easily, just draw three circles that pass through the centers of each other, and the region in the middle is a Reuleaux triangle as shown here:

You can make these with 5, 7, 9 or any odd number of segments by constructing arcs on regular polygons.

George Hart made these solids of uniform width. "Reuleaux acorns" he calls them. They're a 3d generalization of Reuleaux triangles - if you place a flat board on top of them they will roll around smoothly and the board won't wobble up and down. Weird...

## 8.6.12

### Shower decoration code

Last week I wrote about embedding a meaning in artwork. Here's an example from my house back in the United States. We tiled the bathroom, and I embedded some of my wife's glass pieces in the cement. There were long squiggly pieces and small round pieces, perfect for dots and dashes to embed words in Morse code. At the head of the shower: -. .- -.-- .-.. --- .-. ...

At the end of the tub the beautifully symmetric: .- -. -. .-

And along the side of the tub: .--. .- -- and -- .. -.- . and .--. . - . .-. and -- .- --. --. .. .

It was fun making patterns with color and shapes and noticing patterns in the dots and dashes. And when it was done, it looked simply like a random design, but closer inspection hints at a structure that just might invite someone to look deeper. None of our visitors ever thought about, but I knew it was there and it make me happy every time.

At the end of the tub the beautifully symmetric: .- -. -. .-

And along the side of the tub: .--. .- -- and -- .. -.- . and .--. . - . .-. and -- .- --. --. .. .

It was fun making patterns with color and shapes and noticing patterns in the dots and dashes. And when it was done, it looked simply like a random design, but closer inspection hints at a structure that just might invite someone to look deeper. None of our visitors ever thought about, but I knew it was there and it make me happy every time.

## 5.6.12

### Apple Pages regular polygon?

I use the drawing tools in Apple Pages quite a bit. But every time I need a regular polygon, I struggle to find it on the shape menu. See why? The icon is the second from the bottom on the drop-down menu:

I simply cannot get myself to see this icon as a regular pentagon! I often select the 'diamond' (canted square) icon by mistake.

This is an interesting shape, though. It looks like it's the bottom half of a regular hexagon with half a square stuck on top of it. Does this shape have any interesting properties? What is the ratio of long edge:short edge? If the short edge is length 1, what are the lengths of the diagonals? These lengths and areas may not play well together; the hexagon is related to √3 while the square is related to √2. It might be fun to investigate anyways. The angles of this shape are 120°, 120°, 90°, 105° and 105°. And the average of these angle is... exactly 108°! The same as the angle measure of a regular pentagon! (Is this a surprise? Should it be a surprise?)

All right, all fun aside, the icon confuses me when I want a regular polygon (which happens more often that you'd think). I submitted feedback to Apple today asking them to change this icon... I'm just doing my job to promote good geometry. Do you use Apple Pages to make drawings? Has this bothered you? Here, tell Apple: http://www.apple.com/feedback/pages.html.

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