3D Analogue of a Koch Snowflake
Learn about the 3D analogue of a Koch Snowflake construction, which results in a cube in the limit. Read this blog to find out more and see screenshots from a Geogebra applet!
Dave Richeson
Mathematician. John J. & Ann Curley Chair in Liberal Arts at Dickinson College. Author of Tales of Impossibility and Euler's Gem. Coffee drinker.
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Today while reading a Martin Gardner article, I read this paragraph—stating something that I did not know. Perform the 3D analogue of a Koch Snowflake construction but start with a tetrahedron. In the limit you end up with... a cube!
— Dave Richeson (@divbyzero) April 17, 2023
Here are some screenshots from a
1/2 pic.twitter.com/f7IjlwQgtB -
Geogebra applet I found online: https://t.co/dWe7O1ebE1
— Dave Richeson (@divbyzero) April 17, 2023
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Here's a follow-up comment based on a question by @RobJLow. I think Martin Gardner's paragraph is correct but a little misleading. The final limiting shape is a cube (I tried creating the infinite series of the sum of the areas and it is a geometric series that converges to
— Dave Richeson (@divbyzero) April 17, 2023
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the volume of the cube).
— Dave Richeson (@divbyzero) April 17, 2023
However, I *think* that the area surface area goes to infinity. The number of faces at stage n is 4*6^(n-1) and area of each face at step n is (sqrt(3)/2)/4^(n-1). So the total surface area at step n is 2sqrt(3)(3/2)^(n-1), which goes to infinity as
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n goes to infinity. In the limit, however, all of these infinite wiggles become the flat faces of the cube and it has the surface area of the cube.
— Dave Richeson (@divbyzero) April 17, 2023
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Here's a simplified version of the applet I posted yesterday https://t.co/yDwpWm7Goz.
— Dave Richeson (@divbyzero) April 18, 2023
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I decided to write some of this up in case I wanted to use it later. It is a nice application of some 3D-geometry, sequences, and geometric series. It still has some unresolved (to me) details about the geometry though.
— Dave Richeson (@divbyzero) April 18, 2023
7/2 pic.twitter.com/nSBmYnWijs