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Earrings: 256-gon on a 16 x 16 Grid (Swirly Version)

Earrings: 256-gon on a 16 x 16 Grid (Swirly Version)

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Overview

  • Handmade item
  • Materials: acrylic, laser cut, brass, hypoallergenic brass
  • Favorited by: 3 people
  • Gift message available
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From Seattle, WA

Description

256-gon on 16x16 Grid (after Unit Circle Inversion)
White Acrylic, laser cut.
1/8" width.
Mounted on black earring hooks, made from hypoallergenic brass.
Seattle, WA

This design is part of the 256-gon collection, which features a variety of geometric transformations of a 256-gon.

Here, we've replaced each straight line segment from our original design with a swirly S shape. This swirly S shape was generated by taking two copies of the graph of r=tcos(t) parametrized, where t is restricted to the domain [0, 3pi/2].
-------------------------------
In the winter of 2014, my friend and colleague Dan at Math for Love posed the following problem to me:

Suppose you make a polygon by connecting dot-to-dot-dot on a 4x4 grid. What is the largest possible number of sides?

I put this question in front of students immediately, within minutes of first hearing it, and before having a chance to try it on my own at all. Within hours, I had enjoyed puzzling with this problem with three different groups of students. The first group was mostly kindergarteners. I was happy that they managed to draw polygons and count the number of sides, although no one got more than 13 sides. The second group found a much more elegant solution: they found 16 on a 4x4. But they weren't convinced this was the best. And the third group managed to rigorously prove that 16 was an ideal solution on 4x4: with 16 dots, you have a maximum of 16 vertices. A polygon has the same number of vertices and sides, so more than 16 sides is out of the question.

When I got home, I generalized this 16-gon to the fractal you see here. By taking 4 copies of the smaller solution, I can easily create a 64-gon on an 8x8 grid, a 256-gon on a 16x6 grid, etc. I also managed to create a 36-gon on a 6x6 grid, which also naturally extends to a 12x12, 24x24, 48x48, etc.

But this left a whole lot of grids untouched! I convinced myself 9 was impossible on the 3x3. But the 5x5, 7x7, 9x9, 10x10, 11x11, 13x13, 14x14, 15x15, ...

So, I did the natural thing: I shared this problem with a group of students in a week long summer camp. The following day, a student came in with "ideal solutions" on the 10x10 and 11x11. Over the next few days, the 7x7,, 9x9, and everything else up to 20x20 was solved. But the 5x5 remained a mystery.

And still, two years later, the 5x5 is a mystery to me. I haven't been able to make a 25-gon. I'm reasonably confident it's impossible to make a 25-gon. But I'd love it if someone could convince me of that...

---------------------------------
Update!
The previous description was written in early 2017, when I first created this page. But as of the summer of 2017, the 5x5 has been proven impossible! Credit to this result is due to my sister Ayla Gafni and her collaborator Sam Chow.
---------------
Design and assembly by Paul Gafni
Fixtures sourced from Etsy: https://www.etsy.com/listing/164008301/16-pcs-black-ear-wires-22g-french-ear?ga_order=most_relevant&ga_search_type=all&ga_view_type=gallery&ga_search_query=black%20ear%20wires%2016%20pcs&ref=sr_gallery_1
Laser cutting sourced from Ponoko.
256-gon on 16x16 Grid (after Unit Circle Inversion)
White Acrylic, laser cut.
1/8" width.
Mounted on black earring hooks, made from hypoallergenic brass.
Seattle, WA

This design is part of the 256-gon collection, which features a variety of geometric transformations of a 256-gon.

Here, we've replaced each straight line segment from our original design with a swirly S shape. This swirly S shape was generated by taking two copies of the graph of r=tcos(t) parametrized, where t is restricted to the domain [0, 3pi/2].
-------------------------------
In the winter of 2014, my friend and colleague Dan at Math for Love posed the following problem to me:

Suppose you make a polygon by connecting dot-to-dot-dot on a 4x4 grid. What is the largest possible number of sides?

I put this question in front of students immediately, within minutes of first hearing it, and before having a chance to try it on my own at all. Within hours, I had enjoyed puzzling with this problem with three different groups of students. The first group was mostly kindergarteners. I was happy that they managed to draw polygons and count the number of sides, although no one got more than 13 sides. The second group found a much more elegant solution: they found 16 on a 4x4. But they weren't convinced this was the best. And the third group managed to rigorously prove that 16 was an ideal solution on 4x4: with 16 dots, you have a maximum of 16 vertices. A polygon has the same number of vertices and sides, so more than 16 sides is out of the question.

When I got home, I generalized this 16-gon to the fractal you see here. By taking 4 copies of the smaller solution, I can easily create a 64-gon on an 8x8 grid, a 256-gon on a 16x6 grid, etc. I also managed to create a 36-gon on a 6x6 grid, which also naturally extends to a 12x12, 24x24, 48x48, etc.

But this left a whole lot of grids untouched! I convinced myself 9 was impossible on the 3x3. But the 5x5, 7x7, 9x9, 10x10, 11x11, 13x13, 14x14, 15x15, ...

So, I did the natural thing: I shared this problem with a group of students in a week long summer camp. The following day, a student came in with "ideal solutions" on the 10x10 and 11x11. Over the next few days, the 7x7,, 9x9, and everything else up to 20x20 was solved. But the 5x5 remained a mystery.

And still, two years later, the 5x5 is a mystery to me. I haven't been able to make a 25-gon. I'm reasonably confident it's impossible to make a 25-gon. But I'd love it if someone could convince me of that...

---------------------------------
Update!
The previous description was written in early 2017, when I first created this page. But as of the summer of 2017, the 5x5 has been proven impossible! Credit to this result is due to my sister Ayla Gafni and her collaborator Sam Chow.
---------------
Design and assembly by Paul Gafni
Fixtures sourced from Etsy: https://www.etsy.com/listing/164008301/16-pcs-black-ear-wires-22g-french-ear?ga_order=most_relevant&ga_search_type=all&ga_view_type=gallery&ga_search_query=black%20ear%20wires%2016%20pcs&ref=sr_gallery_1
Laser cutting sourced from Ponoko.

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MathematicalMarvels made this item with help from

  • Ponoko, Oakland, CA
  • TheJewelryWorkroom, United States
MathematicalMarvels made this item with help from:
  • Ponoko, Oakland, CA
  • TheJewelryWorkroom, United States

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