Re: [math-fun] Tantalizing Mandelflake fill
On Thu, Sep 3, 2020 at 4:56 PM Bill Gosper <billgosper@gmail.com> wrote:
What the heck is going on here? Alex's (fill rate) smoother idea on Mandelbrot's
Snowflake filler: [Picture <http://gosper.org/smoothman9colors.png>, worth a thousand words, censored for bagbitting math-fun.]
somehow manages to fill both a Snowflake and one √3 times larger with the exact same pattern. Mandelbrot missed it, along with everybody else. I never suspected it was even possible. What's the rule? Note that successive sizes are "texturally everted". —Bill
Julian to the rescue. <http://gosper.org/smoothflake.png> Oh what I'd give for BBM's reaction to this! Speaking of bounding boxes|circumradii|convex hulls, recall APG's heroic extraction of a gross of digits of the circumradius of the France ("Flowsnake") Fractal. For which we never found an expression. Do we even have an accurate bounding box for the terDragon? I just approximated (minus) the left bound of Alex Roodman's i √7 spacefiller <http://gosper.org/smoothman.png> as FromContinuedFraction{0, 18, 29, 2, 11, 1, 1, 1, 3, 2, 5, 2, 94, 1, 1, 1, 42, 1, 2, 1, 2, 1, 1, 19, 23, 1, 2, 1, 2, 16, 55, 1, 14, 93, 1, 23, 1, 2}. Octupling the sampling frequency gave {0, 18, 29, 2, 11, 1, 1, 1, 3, 2, 27, 2, 1, 2, 2, 2, 1, 2, 34, 12, 1, 8, 1, 4, 3, 1, 1, 2, 1, 1, 1, 3, 24, 1, 3, 10, 2, 1, 1, 10, 1, 1, 1, 1, 1, 1, 18, 1, 1, 1, 3, 3} I see no sign of √7. I always rave about piecewiserecursivefractal's inversion capability, but I
should caution that fractal functions like the (optionally) smooth flakefill have inverses in finite terms only on arguments of the form a + i b √3, where a and b are rational.
—rwg
Bill, That curvy Snowflake-filler is nice! Here's something I wrote aboutJörgArndt's solution for filling the Koch snowflake, where the segments of the curve all have the same length: https://spacefillingcurves.wordpress.com/2019/09/01/on-filling-the-koch-snow... -j ‐‐‐‐‐‐‐ Original Message ‐‐‐‐‐‐‐ On Friday, September 4, 2020 9:38 PM, Bill Gosper <billgosper@gmail.com> wrote:
On Thu, Sep 3, 2020 at 4:56 PM Bill Gosper <billgosper@gmail.com> wrote:
What the heck is going on here? Alex's (fill rate) smoother idea on Mandelbrot's
Snowflake filler: [[Picture](http://gosper.org/smoothman9colors.png), worth a thousand words, censored for bagbitting math-fun.]
somehow manages to fill both a Snowflake and one √3 times larger with the exact same pattern. Mandelbrot missed it, along with everybody else. I never suspected it was even possible. What's the rule? Note that successive sizes are "texturally everted". —Bill
[Julian to the rescue.](http://gosper.org/smoothflake.png) Oh what I'd give for BBM's reaction to this! Speaking of bounding boxes|circumradii|convex hulls, recall APG's heroic extraction of a gross of digits of the circumradius of the France ("Flowsnake") Fractal. For which we never found an expression. Do we even have an accurate bounding box for the terDragon? I just approximated (minus) the left bound of [Alex Roodman's i √7 spacefiller](http://gosper.org/smoothman.png) as FromContinuedFraction{0, 18, 29, 2, 11, 1, 1, 1, 3, 2, 5, 2, 94, 1, 1, 1, 42, 1, 2, 1, 2, 1, 1, 19, 23, 1, 2, 1, 2, 16, 55, 1, 14, 93, 1, 23, 1, 2}. Octupling the sampling frequency gave
{0, 18, 29, 2, 11, 1, 1, 1, 3, 2, 27, 2, 1, 2, 2, 2, 1, 2, 34, 12, 1, 8, 1, 4, 3, 1, 1, 2, 1, 1, 1, 3, 24, 1, 3, 10, 2, 1, 1, 10, 1, 1, 1, 1, 1, 1, 18, 1, 1, 1, 3, 3}
I see no sign of √7.
I always rave about piecewiserecursivefractal's inversion capability, but I should caution that fractal functions like the (optionally) smooth flakefill have inverses in finite terms only on arguments of the form a + i b √3, where a and b are rational.
—rwg
On Fri, Sep 4, 2020 at 9:38 PM Bill Gosper <billgosper@gmail.com> wrote:
On Thu, Sep 3, 2020 at 4:56 PM Bill Gosper <billgosper@gmail.com> wrote:
What the heck is going on here? Alex's (fill rate) smoother idea on Mandelbrot's
Snowflake filler: [Picture <http://gosper.org/smoothman9colors.png>, worth a thousand words, censored for bagbitting math-fun.]
somehow manages to fill both a Snowflake and one √3 times larger with the exact same pattern. Mandelbrot missed it, along with everybody else. I never suspected it was even possible. What's the rule? Note that successive sizes are "texturally everted". —Bill
Julian to the rescue. <http://gosper.org/smoothflake.png> Oh what I'd give for BBM's reaction to this!
Speaking of bounding boxes|circumradii|convex hulls, recall APG's heroic extraction of a gross of digits of the circumradius of the France ("Flowsnake") Fractal. For which we never found an expression. Do we even have an accurate bounding box for the terDragon?
Apparently we didn't! I compute it to be {{-1/16, 17/16}, ±3√3 I/16} ={{Re[terDragon[1/240],Re[terDragon[239/240]]}, ±Im[terDragon[t]]} (another 239 factoid!) where t = precisely 113022934338503101/337661456635224360 = FromContinuedFraction@{0, 2, 1, 79, 3, 4, 3, 1, 1, 3, 2, 1, 1, 1, 1, 5905, 5, 2, 1, 2, 2, 3, 2, 74, 3, 2, 37, 1, 2} ! terDragon@t = 299201287/573956280 + 3 I √3/16 which is why I think I haven't slipped off the rails here. But it's admittedly hard to imagine why Im[terDragon] should be maximal at such an exotic abscissa. —rwg If you want to mess with this stuff, especially if you want to fact-check my crazy value of t, ask me to send you Julian's relevant magicware. I just approximated (minus) the left bound of Alex Roodman's i √7
spacefiller <http://gosper.org/smoothman.png> as FromContinuedFraction{0, 18, 29, 2, 11, 1, 1, 1, 3, 2, 5, 2, 94, 1, 1, 1, 42, 1, 2, 1, 2, 1, 1, 19, 23, 1, 2, 1, 2, 16, 55, 1, 14, 93, 1, 23, 1, 2}. Octupling the sampling frequency gave
{0, 18, 29, 2, 11, 1, 1, 1, 3, 2, 27, 2, 1, 2, 2, 2, 1, 2, 34, 12, 1, 8, 1, 4, 3, 1, 1, 2, 1, 1, 1, 3, 24, 1, 3, 10, 2, 1, 1, 10, 1, 1, 1, 1, 1, 1, 18, 1, 1, 1, 3, 3}
I see no sign of √7.
I always rave about piecewiserecursivefractal's inversion capability, but
I should caution that fractal functions like the (optionally) smooth flakefill have inverses in finite terms only on arguments of the form a + i b √3, where a and b are rational.
—rwg
I wasn't completely off the rails. Just out at the end of a very obscure spur. Julian points out that the terDragon "north pole" is flat on top! "The preimage of the top boundary is all points whose base-81 expansions consist of 9s and 27s (i.e. the ternary expansion has blocks of 1000 and 0100). The simplest preimages are 9/80 and 27/80." I.e. In[367]:= trydrag /@ ({9, 27}/80) Out[367]= {{3/16 + (3 I Sqrt[3])/16}, {9/16 + (3 I Sqrt[3])/16}} Here's a crude polygonal closeup of the north polar region <http://gosper.org/tdragpole.png>, which could be at any sufficiently high magnification. Each of the four uppermost prongs is actually a lo-res (irresolute?) representation of the whole patch, and touches the "ceiling" in a Cantor set. Here's a higher-res view <http://gosper.org/tdragwholepole.png> including the whole 9/80 - 27/80 interval. Which is most of the north coast <http://gosper.org/tdragmed.png>! —rwg PS, Julian points out that there are other tangents to the terDragon (boundary) that intersect it in non-Cantor infinite sets. At angles "atan(23/9√3) plus multiples of π/6". On Sat, Sep 5, 2020 at 2:45 PM Bill Gosper <billgosper@gmail.com> wrote:
On Fri, Sep 4, 2020 at 9:38 PM Bill Gosper <billgosper@gmail.com> wrote:
On Thu, Sep 3, 2020 at 4:56 PM Bill Gosper <billgosper@gmail.com> wrote:
What the heck is going on here? Alex's (fill rate) smoother idea on Mandelbrot's
Snowflake filler: [Picture <http://gosper.org/smoothman9colors.png>, worth a thousand words, censored for bagbitting math-fun.]
somehow manages to fill both a Snowflake and one √3 times larger with the exact same pattern. Mandelbrot missed it, along with everybody else. I never suspected it was even possible. What's the rule? Note that successive sizes are "texturally everted". —Bill
Julian to the rescue. <http://gosper.org/smoothflake.png> Oh what I'd give for BBM's reaction to this!
Speaking of bounding boxes|circumradii|convex hulls, recall APG's heroic extraction of a gross of digits of the circumradius of the France ("Flowsnake") Fractal. For which we never found an expression. Do we even have an accurate bounding box for the terDragon?
Apparently we didn't! I compute it to be
{{-1/16, 17/16}, ±3√3 I/16}
={{Re[terDragon[1/240],Re[terDragon[239/240]]}, ±Im[terDragon[t]]}
(another 239 factoid!) where
t = precisely 113022934338503101/337661456635224360 = FromContinuedFraction@{0, 2, 1, 79, 3, 4, 3, 1, 1, 3, 2, 1, 1, 1, 1, 5905, 5, 2, 1, 2, 2, 3, 2, 74, 3, 2, 37, 1, 2} !
terDragon@t = 299201287/573956280 + 3 I √3/16
which is why I think I haven't slipped off the rails here. But it's admittedly hard to imagine why Im[terDragon] should be maximal at such an exotic abscissa.
It's because it's maximal at an infinite Cantor set of abscissas. "North pole" is utterly misleading.
—rwg If you want to mess with this stuff, especially if you want to fact-check my crazy value of t, ask me to send you Julian's relevant magicware.
I just approximated (minus) the left bound of Alex Roodman's i √7
spacefiller <http://gosper.org/smoothman.png> as FromContinuedFraction{0, 18, 29, 2, 11, 1, 1, 1, 3, 2, 5, 2, 94, 1, 1, 1, 42, 1, 2, 1, 2, 1, 1, 19, 23, 1, 2, 1, 2, 16, 55, 1, 14, 93, 1, 23, 1, 2}. Octupling the sampling frequency gave
{0, 18, 29, 2, 11, 1, 1, 1, 3, 2, 27, 2, 1, 2, 2, 2, 1, 2, 34, 12, 1, 8, 1, 4, 3, 1, 1, 2, 1, 1, 1, 3, 24, 1, 3, 10, 2, 1, 1, 10, 1, 1, 1, 1, 1, 1, 18, 1, 1, 1, 3, 3}
I see no sign of √7.
I always rave about piecewiserecursivefractal's inversion capability, but
I should caution that fractal functions like the (optionally) smooth flakefill have inverses in finite terms only on arguments of the form a + i b √3, where a and b are rational.
—rwg
Notice from Julian: "I need to issue a correction. I derped in the derivation, and that line is not the tangent. The only tangent lines to the terdragon are at multiples of π/6 and at angle -ArcTan[(2 Sqrt[3])/5], and the latter are only tangent at two points. The convex hull has vertices terdrag/@{1/720, 1/240, 1/80, 3/80, 9/80, 27/80, 83/240, 719/720, 239/240, 79/80, 77/80, 71/80, 53/80, 157/240}. Julian" terDragon convex hull <http://gosper.org/terhull.png> (a tetradecagon!) —rwg On Sun, Sep 6, 2020 at 10:30 AM Bill Gosper <billgosper@gmail.com> wrote:
I wasn't completely off the rails. Just out at the end of a very obscure spur. Julian points out that the terDragon "north pole" is flat on top! "The preimage of the top boundary is all points whose base-81 expansions consist of 9s and 27s (i.e. the ternary expansion has blocks of 1000 and 0100). The simplest preimages are 9/80 and 27/80."
I.e. In[367]:= trydrag /@ ({9, 27}/80)
Out[367]= {{3/16 + (3 I Sqrt[3])/16}, {9/16 + (3 I Sqrt[3])/16}}
Here's a crude polygonal closeup of the north polar region <http://gosper.org/tdragpole.png>, which could be at any sufficiently high magnification. Each of the four uppermost prongs is actually a lo-res (irresolute?) representation of the whole patch, and touches the "ceiling" in a Cantor set. Here's a higher-res view <http://gosper.org/tdragwholepole.png> including the whole 9/80 - 27/80 interval. Which is most of the north coast <http://gosper.org/tdragmed.png>! —rwg PS, Julian points out that there are other tangents to the terDragon (boundary) that intersect it in non-Cantor infinite sets. At angles "atan(23/9√3) plus multiples of π/6".
On Sat, Sep 5, 2020 at 2:45 PM Bill Gosper <billgosper@gmail.com> wrote:
On Fri, Sep 4, 2020 at 9:38 PM Bill Gosper <billgosper@gmail.com> wrote:
On Thu, Sep 3, 2020 at 4:56 PM Bill Gosper <billgosper@gmail.com> wrote:
What the heck is going on here? Alex's (fill rate) smoother idea on Mandelbrot's
Snowflake filler: [Picture <http://gosper.org/smoothman9colors.png>, worth a thousand words, censored for bagbitting math-fun.]
somehow manages to fill both a Snowflake and one √3 times larger with the exact same pattern. Mandelbrot missed it, along with everybody else. I never suspected it was even possible. What's the rule? Note that successive sizes are "texturally everted". —Bill
Julian to the rescue. <http://gosper.org/smoothflake.png> Oh what I'd give for BBM's reaction to this!
Speaking of bounding boxes|circumradii|convex hulls, recall APG's heroic extraction of a gross of digits of the circumradius of the France ("Flowsnake") Fractal. For which we never found an expression. Do we even have an accurate bounding box for the terDragon?
Apparently we didn't! I compute it to be
{{-1/16, 17/16}, ±3√3 I/16}
={{Re[terDragon[1/240],Re[terDragon[239/240]]}, ±Im[terDragon[t]]}
(another 239 factoid!) where
t = precisely 113022934338503101/337661456635224360 = FromContinuedFraction@{0, 2, 1, 79, 3, 4, 3, 1, 1, 3, 2, 1, 1, 1, 1, 5905, 5, 2, 1, 2, 2, 3, 2, 74, 3, 2, 37, 1, 2} !
terDragon@t = 299201287/573956280 + 3 I √3/16
which is why I think I haven't slipped off the rails here. But it's admittedly hard to imagine why Im[terDragon] should be maximal at such an exotic abscissa.
It's because it's maximal at an infinite Cantor set of abscissas. "North pole" is utterly misleading.
As if there were finite Cantor sets.
—rwg
If you want to mess with this stuff, especially if you want to fact-check my crazy value of t, ask me to send you Julian's relevant magicware.
I just approximated (minus) the left bound of Alex Roodman's i √7
spacefiller <http://gosper.org/smoothman.png> as FromContinuedFraction{0, 18, 29, 2, 11, 1, 1, 1, 3, 2, 5, 2, 94, 1, 1, 1, 42, 1, 2, 1, 2, 1, 1, 19, 23, 1, 2, 1, 2, 16, 55, 1, 14, 93, 1, 23, 1, 2}. Octupling the sampling frequency gave
{0, 18, 29, 2, 11, 1, 1, 1, 3, 2, 27, 2, 1, 2, 2, 2, 1, 2, 34, 12, 1, 8, 1, 4, 3, 1, 1, 2, 1, 1, 1, 3, 24, 1, 3, 10, 2, 1, 1, 10, 1, 1, 1, 1, 1, 1, 18, 1, 1, 1, 3, 3}
I see no sign of √7.
I always rave about piecewiserecursivefractal's inversion capability,
but I should caution that fractal functions like the (optionally) smooth flakefill have inverses in finite terms only on arguments of the form a + i b √3, where a and b are rational.
—rwg
But they will always have inverses, which can be arbitrarily well-approximated via rational a and b. Otherwise the flakefill wouldn't be a FILL. The inverses are unique if a or b approaches an irrational.
On Mon, Sep 7, 2020 at 10:07 AM Bill Gosper <billgosper@gmail.com> wrote:
Notice from Julian: "I need to issue a correction. I derped in the derivation, and that line is not the tangent. The only tangent lines to the terdragon are at multiples of π/6 and at angle -ArcTan[(2 Sqrt[3])/5], and the latter are only tangent at two points. The convex hull has vertices terdrag/@{1/720, 1/240, 1/80, 3/80, 9/80, 27/80, 83/240, 719/720, 239/240, 79/80, 77/80, 71/80, 53/80, 157/240}.
Julian" terDragon convex hull <http://gosper.org/terhull.png> (a tetradecagon!) —rwg
This black-on-green depiction is actually the fairly convincing optical illusion that the tetradecagonal convex hull is visibly rounded. The illusion dissipates if we replace black with white and green with black <http://gosper.org/terhullbw.png>. I'm not sure if this entirely due to the poor contrast between green and white, vs the black curve being somehow disruptive. Without it, the green looks polygonal.
On Sun, Sep 6, 2020 at 10:30 AM Bill Gosper <billgosper@gmail.com> wrote:
I wasn't completely off the rails. Just out at the end of a very obscure spur. Julian points out that the terDragon "north pole" is flat on top! "The preimage of the top boundary is all points whose base-81 expansions consist of 9s and 27s (i.e. the ternary expansion has blocks of 1000 and 0100). The simplest preimages are 9/80 and 27/80."
I.e. In[367]:= trydrag /@ ({9, 27}/80)
Out[367]= {{3/16 + (3 I Sqrt[3])/16}, {9/16 + (3 I Sqrt[3])/16}}
Here's a crude polygonal closeup of the north polar region <http://gosper.org/tdragpole.png>, which could be at any sufficiently high magnification. Each of the four uppermost prongs is actually a lo-res (irresolute?) representation of the whole patch, and touches the "ceiling" in a Cantor set. Here's a higher-res view <http://gosper.org/tdragwholepole.png> including the whole 9/80 - 27/80 interval. Which is most of the north coast <http://gosper.org/tdragmed.png>! —rwg PS, Julian points out that there are other tangents to the terDragon (boundary) that intersect it in non-Cantor infinite sets. At angles "atan(23/9√3) plus multiples of π/6".
On Sat, Sep 5, 2020 at 2:45 PM Bill Gosper <billgosper@gmail.com> wrote:
On Fri, Sep 4, 2020 at 9:38 PM Bill Gosper <billgosper@gmail.com> wrote:
On Thu, Sep 3, 2020 at 4:56 PM Bill Gosper <billgosper@gmail.com> wrote:
What the heck is going on here? Alex's (fill rate) smoother idea on Mandelbrot's
Snowflake filler: [Picture <http://gosper.org/smoothman9colors.png>, worth a thousand words, censored for bagbitting math-fun.]
somehow manages to fill both a Snowflake and one √3 times larger with the exact same pattern. Mandelbrot missed it, along with everybody else. I never suspected it was even possible. What's the rule? Note that successive sizes are "texturally everted". —Bill
Julian to the rescue. <http://gosper.org/smoothflake.png> Oh what I'd give for BBM's reaction to this!
Speaking of bounding boxes|circumradii|convex hulls, recall APG's heroic extraction of a gross of digits of the circumradius of the France ("Flowsnake") Fractal. For which we never found an expression. Do we even have an accurate bounding box for the terDragon?
Apparently we didn't! I compute it to be
{{-1/16, 17/16}, ±3√3 I/16}
={{Re[terDragon[1/240],Re[terDragon[239/240]]}, ±Im[terDragon[t]]}
(another 239 factoid!) where
t = precisely 113022934338503101/337661456635224360 = FromContinuedFraction@{0, 2, 1, 79, 3, 4, 3, 1, 1, 3, 2, 1, 1, 1, 1, 5905, 5, 2, 1, 2, 2, 3, 2, 74, 3, 2, 37, 1, 2} !
terDragon@t = 299201287/573956280 + 3 I √3/16
which is why I think I haven't slipped off the rails here. But it's admittedly hard to imagine why Im[terDragon] should be maximal at such an exotic abscissa.
It's because it's maximal at an infinite Cantor set of abscissas. "North pole" is utterly misleading.
As if there were finite Cantor sets.
—rwg
If you want to mess with this stuff, especially if you want to fact-check my crazy value of t, ask me to send you Julian's relevant magicware.
I just approximated (minus) the left bound of Alex Roodman's i √7
spacefiller <http://gosper.org/smoothman.png> as FromContinuedFraction{0, 18, 29, 2, 11, 1, 1, 1, 3, 2, 5, 2, 94, 1, 1, 1, 42, 1, 2, 1, 2, 1, 1, 19, 23, 1, 2, 1, 2, 16, 55, 1, 14, 93, 1, 23, 1, 2}. Octupling the sampling frequency gave
{0, 18, 29, 2, 11, 1, 1, 1, 3, 2, 27, 2, 1, 2, 2, 2, 1, 2, 34, 12, 1, 8, 1, 4, 3, 1, 1, 2, 1, 1, 1, 3, 24, 1, 3, 10, 2, 1, 1, 10, 1, 1, 1, 1, 1, 1, 18, 1, 1, 1, 3, 3}
I see no sign of √7.
I always rave about piecewiserecursivefractal's inversion capability,
but I should caution that fractal functions like the (optionally) smooth flakefill have inverses in finite terms only on arguments of the form a + i b √3, where a and b are rational.
—rwg
But they will always have inverses, which can be arbitrarily well-approximated via rational a and b. Otherwise the flakefill wouldn't be a FILL. The inverses are unique if a or b approaches an irrational.
participants (2)
-
Bill Gosper -
Ventrella