Andrew Hoyer, Cameron Browne, & Sphere Inversion

Welcome to this week’s Math Munch! Fractals, origami, math art, games, and a mind-bending video are all ahead, so let’s get into it.

Andrew Hoyer

Andrew Hoyer

First up, let’s take a look at the work of Andrew Hoyer.  According to his website, he’s a “software engineer in his mid-twenties living it up in sometimes sunny San Francisco.” I came across his work when I found his beautiful and completely engaging introduction to simple fractals.  (Go on! Click. Then read, experiment and play!)

Cantor Set

A Cantor set

At the bottom of that page, Andrew links to a wonderful, long list of fractals, arranged by Hausdorff dimension, which is a way of measuring fractals as being something like 2.5 dimensions.  A line is 1 dimensional.  A plane is 2D, and you can find many fractals with dimension in between!!  Weird, right?

I was also really pleased to find Andrew’s Instagram feed, which features some of his beautiful origami creations.  Andrew’s agreed to answer your questions for an upcoming Q&A, so ask away!

Compound of 5 tetrahedra Truncated Icosahedron Cube Dodecahedron
Cameron Browne

Cameron Browne

Up next, meet Cameron Browne. He’s an accomplished researcher who designs and studies games. Take a look at the many many games Cameron has created. The rules and descriptions are there, and Cameron sent along links to playable versions of a few, which you can find by clicking the pictures below.  For the third one, you’ll need to search for “Margo” or “Spargo.” For his research, Cameron investigates the possibilities of artificial intelligence, and how a computer can be used to generate games and puzzles.



Margo and Spargo

Margo and Spargo

Cameron is also an artist, and he has a page full of his graphic designs.  I found Cameron through his page of Truchet curves.  I love the way his pages are full of diagrams and just enough information to start making sense of things, even if it’s not perfectly clear.  Cameron also has MANY pages of wonderful fractal-ish graphics: Impossible Fractals, Cantor Knots, Fractal Board Games, Woven Horns, Efficient Trees, and on and on…  And he has agreed to do a Q&A with us, so please, submit a question. What are you wondering?

A Cantor Knot

A Cantor Knot

A Truchet curve "Mona Lisa"

A Truchet curve “Mona Lisa”

An "impossible" fractal

An “impossible” fractal

And, as if that wasn’t enough mathy awesomeness, check out this video about turning a sphere inside out.  A bit of personal history, I actually used this video  (though it was only on VHS back then, checked out from the library) as part of the research for my independent research project during my senior year of college.  It gets pretty tricky, but if you watch it all the way through it starts to make some sense.

Have a great week.  Bon appetit!

Reflection sheet – Andrew Hoyer, Cameron Browne, & Sphere Inversion

55 responses »

  1. I really liked this video. I thought it was really interesting and a little confusing at the same time. I learned something new today because i thought that you couldn’t take a sphere and turn it inside out. You can by twisting,turning and flipping it in different ways.

    • Hi Cennedy. I felt the same way watching this video! I didn’t know about turning numbers of curves, and smiles, bowls, and saddles of spheres, and how these are their fundamental properties. I really like the graphics too. Things got complicated quickly with the sphere, and I couldn’t always follow them, but I certainly appreciate the ones that I could. YAY! We both learned something really cool today.

  2. This video puzzled me a lot. I couldn’t really understand what was going on, the twists and turns and curves really intrigued me. How can a sphere be turned inside out? I have never asked myself this question until I started watching this wonderful video containing science, geometry, and your brain. I have never been on this website before, but I now know I will DEFINITELY be here many times in the very near future. (:

    • Hey Shelby-
      Turning a sphere inside out really is an interesting idea. It almost makes it weird that you can’t turn a circle inside out.

      I also like the way you said “I have never asked myself this question…” A lot of math can happen when you just try to come up with good questions.

      I’m so glad to hear you’ll be visiting the site again! See you soon, and thank you for commenting.

  3. That was an amazing video! I was like at the beginning this is gonna be so easy but once Cameron said the rules……I was like this is gonna be impossible. It pushed my brain to so many new levels that i never even knew i had but I still could not figure it out. As the video progressed, as she was explaining many of the physics in it I was starting to get a little more confident as she talked about the smiles,frownes,turning numbers, and all of the twists, turns, and curves without any creases or sharp turns either. As the video was ending we got to see how all of the twists and turns worked, and amazingly it worked! It was probably the most interesting video iv’e ever seen in my life!But i doubt its the best because theres so many videos you cant even count on math munch! Thank you guys so much it was so fascinating, and im definetely gonna be sharing this with my family, im gonna tell my cousins and other family members to go onto math munch! Thank you guys so much and this is only the beginning so you better be ready world!

    • Daniel- I’m so glad you loved the sphere inversion video!! It’s really wonderful to have you share your enthusiasm. (One correction, though. Cameron Browne didn’t make the video. I found it somewhere else.) This may have been the best video you’ve ever seen in your life, but for me, this comment was the best I’ve ever read in my life. 🙂 Thanks.

    • Hi Marina! If a 3-D figure has edges—like, for instance, a cube—then it already has the kind of “creases” that are discussed in the video. So I don’t this challenge would work with a cube.

      I think it would work just fine with any sort of 3-D blob shape without any edges, because you could first mold it into a sphere, since the special material is stretchy and flexible. But I’m not sure about more complicated 3-D surfaces, like a donut (aka a torus).

      So I did a little research and found this video through this mathoverflow post. It shows how to evert a torus without creasing it. “Evert” is the technical mathematical term for turning inside-out. Apparently any closed 3-D curvy figure can be everted, no matter how many donut holes it has.

      Thanks for sharing your question, Marina! It helped me to learn some new math. I hope my answer helped you to learn some new math, too.

  4. HI, I just wanted to say that I really love the three pictures on the top of the page that Andrew Hoyer has created. They are really beautiful. I am wondering how he did it but they just caught my eye.

  5. Hi when I saw the video I was inspired with math munch I love your guys work and before I went on math munch math was boring but once I went on math munch I loved math so thanks you very much because of your amazing work I LOVE MATH!!!!!!!!

  6. I really enjoyed the video and would love to see more. like other people it inspired me. Thanks for posting this video. This site makes me like math a little bit more. Most math sites are boring but this one is the best.

  7. That was very intersting and complicated. How did you find this video.? It would be cool if they made paper like that. Imagine how many puzzles you could do with it.

    • Hey Ryan-
      I actually found this video in the library at Lawrence University (where I went to college). It was on a physical tape cassette. One day looking at math videos on youtube, it came up in the suggestions bar!

  8. Wow! That was very complicated for me, I mean to turn a sphere inside out! But then when she explained how, it all made sense! I’m so going to see more videos like this!

    • Some of it was complicated for me to understand too, Gabi. But hopefully that shouldn’t turn us off to something new and really cool that we didn’t think of before.

  9. It was very interesting how you can turn a sphere insideout without making creases.A circle it is impossible to turn insideout.Where do people try that?

  10. This video was very puzzling in the beginning. In the beginning I thought It was impossible to turn a sphere inside out but it turns out I was wrong. I love how they compared it to a monorail. Overall I think this video was great the best part was when they showed how turning a sphere worked and they explained it section by section, making it easier to understand. Could you do that in real life?

  11. Do you know how long it took to devise at least one of the ways to turn the sphere? And when was the first way thought of on record? I found this video very intriguing to watch and see how something that verbally seems so simple can become so complicated on paper.

  12. this video was absolutely amazing! my face was glued to the screen for all of the math filled twenty awesome minutes. though it was a bit hard to understand it, it was nonetheless fun to watch and I feel that I have learned so much! I also think that having the video being hard to understand actually added to all the fun. I felt like I was STRUGGLING to understand it!:)

  13. I really enjoyed this video. The format in which the presented the information was really interesting and helped me understand the way the circle and the sphere were different. I think that I will remeber the information I learned from it for a long time.

  14. I really like this video it is Interesting how you figure out how a turn that is turning to the right, the turning number is -3. And if there is no turning the
    turning number is 0. I really appreciate all the work you guys are doing to make this websiteexist.

  15. This video was completely intriguing!! I was so focused I watched all 21 minutes. I liked how if you create wavy ripples on the surface, you can turn a sphere inside-out, but, you can’t turn a circle inside-out according to the rules that apply. The main one is no sharp angles. I am so curious where you found this amazing video, and when the video was produced, the graphics don’t look that new so I am guessing about 6 years ago?

  16. This video took me a couple minutes to fully understand, but once I did I really enjoyed it! I’ve never considered turning a sphere inside out, but now I can say that know how to. I’m really glad you posted this video because out of all the videos I’ve seen on Math Munch so far, this has to be the most amazing. 🙂

  17. The video is really interesting, but I still don’t understand how to turn a sphere inside out. Wouldn’t twisting it the way they showed in the video make creases?

  18. I really liked this video, its really interesting and confusing. All the twisting, turning, flipping, smile and frown faces puzzled me. When Cameron said the rules I thought it was going to be impossible, but at the end this video showed me how all the turns, twists, and flips worked. It also helped me comprehend how a circle and sphere could be so different.

  19. Wow that was so cool but it was hard understand at first but more I to the video it becomes simple.It seemed almost impossible do to at first but it explains how to do it.

  20. When I watched this Video i was confused on what was happening in my head i thought WHAT IS HAPPENING HERE ! But then when they put the train on the circle i somewhat understood what was happening when they brought up the turning numbers . They helped me understand them with the sad faces and happy faces . I really wish I had paper that could pass through itself !! Thank you for making this video I will watch math munch more often !

  21. WOW!! I’m speechless….that was so cool!!!! I never knew it was possible to turn a sphere inside out!! I like the part in the video where you solve a simpler problem with the circle and its turning number. It eventually helped to see the waves on the circle undo the loops made by the circle (the figure eight). It made me realize that a circle and a sphere TOTALLY different and also I learned something new, which is good!! Thanks for the video, I think I was a little confused so right now I’m going to watch it again! Thanks again!! 🙂

  22. The circle is the integers of rational numbers and bending the circle only corrupts the system to make them strange irrational numbers. I still want to figure out how to make the sphere inside out but you have show me the ropes for a days project. Thank you!

  23. cool but its confusing I think they can explain what it means with more than example like a street with a car but other than that it was great I like how they explained it with things like smiley faces and frown faces!

  24. when this video ended i didnot want it to end !!!! i never new you can take a shere inside out this video made me relize in the world there are many different kinds of shapes.there many different rule that apply to turning a shape in side out it took me while to comprehend this video over all had many loops and turns but it was outstanding

  25. Hi math munch team, i just wanted to say that i really love all of your posts. I think so far this must be my favorite video i ever watched on math munch and any other internet videos. I liked this video a lot beacuse it showed a lot of cool things. Some of the cool things were the bright colors, cool spheres and circles, and the way they had explained to fold a sphere inside out. I also thought that the paper stuff they were explaing was also very cool beacse you could not fold it, bend it, or twist it. Even thought i thought this video was very cool i thought it was also very confusing and difficult to follow. I think they were going a bit to fast for me.Anyway i am going to share this with my family. My dad especially beacse he loves math and now so do i all thanks to you all at math munch. i love math!

  26. i was so amazed on how you cant make a inside out circle but a inside out SPHERE and the monorail was really cool but i had a little trouble following

  27. I never even thought about finding fractals in my blood vessels! Math is everywhere! All of the different examples (cantor, Pythagoras tree, Koch curve, H-tree, Sierpinski Carpet) were really cool. My favorite one to hover the mouse over was the Pythagores Tree. Math can be a cool piece of art! The H-tree was a little hard to see the 180 degree angle, but I got it!

  28. In the beginning when the women said, well let me show you and then proceeded to work out all of those complicated steps i just thought, well this isn’t going to end well for me. Then they started using all those helpful analogies and it didn’t totally click ,but it made much more sense.The belt and end examples are what really made sense to me.

  29. That was one of the most interesting videos I have ever seen. Although it was very hard to understand in the beggining I started to understand it better in the beggining and end but I think that I figured it out.

  30. This is now by far the most interesting math video I have ever watched. If someone where to ask you if you could turn a sphere inside out, without cutting or creasing it, you would say no. But after watching this video you can picture how you could by altering the rules just slightly. I loved the discussion of how inverting spheres compares to inverting circles!

  31. This video was very puzzling in the beginning. In the beginning I thought It was impossible to turn a sphere inside out but it turns out I was wrong. I wonder how many steps it took to actually turn the sphere inside out

  32. Pingback: Zippergons, High Fashion, and Really Big Numbers | Math Munch

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