Most of you know that it makes a cube: gosper.org/3dsno.png at least if you unite closed solids. The sequence of boundaries converges to a fractal that so distracted Mandelbrot that he neglected to mention its cubicalness. If each octant of this fractal is everted, so that all eight corners wind up in the center, we do not see eight "corner reflectors", but rather a hint of what BBM was trying to describe. gosper.org/unstella.svg except that the little octahedra should be little gobs of razor blades. The true limiting fractal here is the disjoint union of six copies of itself placed on the vertices of its bounding octahedron, and then scaled by 1/2. The intersection of this figure with its bounding octahedron is eight Sierpinski gaskets. --rwg Metallic instances of the "eight corner reflectors" figure are hung from boat masts as radar reflectors.
Maybe it's the yellow color, but this everted fractal reminds me perforce of a durian fruit. —Dan
On Mar 16, 2016, at 6:42 PM, Bill Gosper <billgosper@gmail.com> wrote:
If each octant of this fractal is everted, so that all eight corners wind up in the center, we do not see eight "corner reflectors", but rather a hint of what BBM was trying to describe. gosper.org/unstella.svg <http://gosper.org/unstella.svg> except that the little octahedra should be little gobs of razor blades. The true limiting fractal here is the disjoint union of six copies of itself placed on the vertices of its bounding octahedron, and then scaled by 1/2. The intersection of this figure with its bounding octahedron is eight Sierpinski gaskets. --rwg
P.S. What happens if at each stage of the 3D snowflake, the new pyramids are derected *inward* instead of being erected outward? The resulting geode could for viewing be sliced open or subjected to inversion in a the circumscribed sphere. —Dan
On Mar 16, 2016, at 11:25 PM, Dan Asimov <asimov@msri.org> wrote:
Maybe it's the yellow color, but this everted fractal reminds me perforce of a durian fruit.
—Dan
On Mar 16, 2016, at 6:42 PM, Bill Gosper <billgosper@gmail.com> wrote:
If each octant of this fractal is everted, so that all eight corners wind up in the center, we do not see eight "corner reflectors", but rather a hint of what BBM was trying to describe. gosper.org/unstella.svg <http://gosper.org/unstella.svg> except that the little octahedra should be little gobs of razor blades. The true limiting fractal here is the disjoint union of six copies of itself placed on the vertices of its bounding octahedron, and then scaled by 1/2. The intersection of this figure with its bounding octahedron is eight Sierpinski gaskets. --rwg
Nice! A cross-section perpendicular to an axis 3/4 of the way along the length gives the "box fractal". It would be fun to see an animation of the slices. On Wed, Mar 16, 2016 at 6:42 PM, Bill Gosper <billgosper@gmail.com> wrote:
Most of you know that it makes a cube: gosper.org/3dsno.png at least if you unite closed solids. The sequence of boundaries converges to a fractal that so distracted Mandelbrot that he neglected to mention its cubicalness. If each octant of this fractal is everted, so that all eight corners wind up in the center, we do not see eight "corner reflectors", but rather a hint of what BBM was trying to describe. gosper.org/unstella.svg except that the little octahedra should be little gobs of razor blades. The true limiting fractal here is the disjoint union of six copies of itself placed on the vertices of its bounding octahedron, and then scaled by 1/2. The intersection of this figure with its bounding octahedron is eight Sierpinski gaskets. --rwg
Metallic instances of the "eight corner reflectors" figure are hung from boat masts as radar reflectors. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
On 2016-03-17 07:07, Mike Stay wrote:
Nice! A cross-section perpendicular to an axis 3/4 of the way along the length gives the "box fractal". It would be fun to see an animation of the slices. In that vein: https://www.simonsfoundation.org/multimedia/mathematical-impressions-the-sur... [1]
Mathematica is developing a "region calculus", e.g., gosper.org/cubeslice.png [2] that might soon facilitate this. --rwg
On Wed, Mar 16, 2016 at 6:42 PM, Bill Gosper <billgosper@gmail.com> wrote:
Most of you know that it makes a cube: gosper.org/3dsno.png at least if you unite closed solids. The sequence of boundaries converges to a fractal that so distracted Mandelbrot that he neglected to mention its cubicalness. If each octant of this fractal is everted, so that all eight corners wind up in the center, we do not see eight "corner reflectors", but rather a hint of what BBM was trying to describe. gosper.org/unstella.svg except that the little octahedra should be little gobs of razor blades. The true limiting fractal here is the disjoint union of six copies of itself placed on the vertices of its bounding octahedron, and then scaled by 1/2. The intersection of this figure with its bounding octahedron is eight Sierpinski gaskets. --rwg
Metallic instances of the "eight corner reflectors" figure are hung from boat masts as radar reflectors. _______________________________________________
Links: ------ [1] https://www.simonsfoundation.org/multimedia/mathematical-impressions-the-sur... [2] http://ma.sdf.org/gosper.org/cubeslice.png
If I understand this correctly, it looks like the surface tetrahedrons touch each other starting at the third iteration, completely sealing off chambers below the surface which continue to be filled in as the iterations progress. And though the surface makes contact with itself, it never actually passes through itself, is that right? Question: Is there a 4D analog to the snowflake curve, based on 4D simplexes? If so, would the surface of this curve pass through itself? Tom Bill Gosper writes:
Most of you know that it makes a cube: gosper.org/3dsno.png at least if you unite closed solids. The sequence of boundaries converges to a fractal that so distracted Mandelbrot that he neglected to mention its cubicalness. If each octant of this fractal is everted, so that all eight corners wind up in the center, we do not see eight "corner reflectors", but rather a hint of what BBM was trying to describe. gosper.org/unstella.svg except that the little octahedra should be little gobs of razor blades. The true limiting fractal here is the disjoint union of six copies of itself placed on the vertices of its bounding octahedron, and then scaled by 1/2. The intersection of this figure with its bounding octahedron is eight Sierpinski gaskets. --rwg
Metallic instances of the "eight corner reflectors" figure are hung from boat masts as radar reflectors. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 2016-03-18 06:21, Tom Karzes wrote:
If I understand this correctly, it looks like the surface tetrahedrons touch each other starting at the third iteration, completely sealing off chambers below the surface which continue to be filled in as the iterations progress. And though the surface makes contact with itself, it never actually passes through itself, is that right?
Yes. The fractal boundary reminds me of the inside of a crab.
Question: Is there a 4D analog to the snowflake curve, based on 4D simplexes? If so, would the surface of this curve pass through itself?
Tom
Sounds plausible, but 4space is very spacious.
George Hart been here, done this eight years ago. --rwg http://www.georgehart.com/ProceduralGeneration/Bridges08-Hart10pages.pdf
Bill Gosper writes:
Most of you know that it makes a cube: gosper.org/3dsno.png at least if you unite closed solids. The sequence of boundaries converges to a fractal that so distracted Mandelbrot that he neglected to mention its cubicalness. If each octant of this fractal is everted, so that all eight corners wind up in the center, we do not see eight "corner reflectors", but rather a hint of what BBM was trying to describe. gosper.org/unstella.svg except that the little octahedra should be little gobs of razor blades. The true limiting fractal here is the disjoint union of six copies of itself placed on the vertices of its bounding octahedron, and then scaled by 1/2. The intersection of this figure with its bounding octahedron is eight Sierpinski gaskets. --rwg
Metallic instances of the "eight corner reflectors" figure are hung from boat masts as radar reflectors.
producing a strong echo while flapping in the breeze.
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participants (5)
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Bill Gosper -
Dan Asimov -
Mike Stay -
rwg -
Tom Karzes