[math-fun] Brain hemisphere spacefills likely with sextuple points.
Julian & I (again mostly Julian) found exactly the two continuous functions schematized by this old Macsyma plot <http://gosper.org/hemis.gif> of a closed and unclosed curve. So now we can sample the functions at arbitrary rationals. E.g., triple frequency <http://gosper.org/semihexes3x.png>. But actually the median curve <http://gosper.org/semihexesmedian.png> most resembles the Macsyma plot. Since Julian recently found a sampling of Heighway's Dragon equal to its median curve, it's likely that there are samplings of these semihexes coincident with their median curves. This will be easy to test when we manage to invert the semihex functions. Which will also let us directly exhibit the sextuple points. Failing that, it should be possible to tediously guess and check a few. (Actually, since the Dragon truly spacefills, there is obviously some way to hit all the vertices on the median curve. But "luckily" not crazy and disordered.) Heretofore, the only spacefill dense with sextuple points I'd seen was Julian's huge Fourier triangle (@http://howwords.com/triangles/artcircle, which seems to be down), made from 25 copies of itself. Unfortunately, sextuple points are visited very furtively, so the Fourier plot is unconvincing, never (in six tries) coming very close to the actual sextuple points. For the exact definition, see my 21 Aug 2016 math-fun, which gives the six preimages of In[381]:= untrifil25[1/2 + I Sqrt[3]/10] Out[381]= {8/75, 11/75, 14/75, 17/75, 4/15, 23/75} (In arithmetic progression. Always?) —rwg
I confess to being too stupid to construct the inversesemihexes function, even with Julian's magic toolbox. But, spazzing around, I found that a hexagon *vertex* is a sextuple point!? In[483]:= Cases[Table[{t, semihexes@t}, {t, 0, 1, 1/64/15}], {_, {1 + I √3/2}}] Out[483]= {{21/80, {1 + (I √3)/2}},{43/160, {1 + (I √3)/2}}, {13/40, {1 + (I √3)/2}}, {53/160, {1 + (I √3)/2}}, {77/160, {1 + (I √3)/2}},{39/80, {1 + (I √3)/2}}} This seems to say that if you extend the curve to make three copies sharing that point, it would be 18-tuple!? I find this incredible. —rwg On Sat, Nov 30, 2019 at 12:20 AM Bill Gosper <billgosper@gmail.com> wrote:
Julian & I (again mostly Julian) found exactly the two continuous functions schematized by this old Macsyma plot <http://gosper.org/hemis.gif> of a closed and unclosed curve. So now we can sample the functions at arbitrary rationals. E.g., triple frequency <http://gosper.org/semihexes3x.png>. But actually the median curve <http://gosper.org/semihexesmedian.png> most resembles the Macsyma plot. Since Julian recently found a sampling of Heighway's Dragon equal to its median curve, it's likely that there are samplings of these semihexes coincident with their median curves. This will be easy to test when we manage to invert the semihex functions. Which will also let us directly exhibit the sextuple points. Failing that, it should be possible to tediously guess and check a few. (Actually, since the Dragon truly spacefills, there is obviously some way to hit all the vertices on the median curve. But "luckily" not crazy and disordered.)
Heretofore, the only spacefill dense with sextuple points I'd seen was Julian's huge Fourier triangle (@http://howwords.com/triangles/artcircle, which seems to be down), made from 25 copies of itself. Unfortunately, sextuple points are visited very furtively, so the Fourier plot is unconvincing, never (in six tries) coming very close to the actual sextuple points. For the exact definition, see my 21 Aug 2016 math-fun, which gives the six preimages of In[381]:= untrifil25[1/2 + I Sqrt[3]/10]
Out[381]= {8/75, 11/75, 14/75, 17/75, 4/15, 23/75} (In arithmetic progression. Always?) —rwg
Argh, I'm an idiot. False alarm: 1+i√3/2 is not a vertex of either semihex! It's in the east quadrant of the closed (upper) loop. Apologies to all who were contemplating a psychiatric intervention. Here are some views in one zoomable frame. <http://gosper.org/sextups.png> Note that three of the six grazing regions resemble daisy-chained horseshoe crabs. Do they look familiar to anyone? (Other than crab voyeurs,) —rwg On Sat, Nov 30, 2019 at 10:07 PM Bill Gosper <billgosper@gmail.com> wrote:
I confess to being too stupid to construct the inversesemihexes function, even with Julian's magic toolbox. But, spazzing around, I found that a hexagon *vertex* is a sextuple point!?
In[483]:= Cases[Table[{t, semihexes@t}, {t, 0, 1, 1/64/15}], {_, {1 + I √3/2}}]
Out[483]=
{{21/80, {1 + (I √3)/2}},{43/160, {1 + (I √3)/2}}, {13/40, {1 + (I √3)/2}},
{53/160, {1 + (I √3)/2}}, {77/160, {1 + (I √3)/2}},{39/80, {1 + (I √3)/2}}}
This seems to say that if you extend the curve to make three copies
sharing that point, it would be 18-tuple!? I find this incredible.
—rwg
On Sat, Nov 30, 2019 at 12:20 AM Bill Gosper <billgosper@gmail.com> wrote:
Julian & I (again mostly Julian) found exactly the two continuous functions schematized by this old Macsyma plot <http://gosper.org/hemis.gif> of a closed and unclosed curve. So now we can sample the functions at arbitrary rationals. E.g., triple frequency <http://gosper.org/semihexes3x.png>. But actually the median curve <http://gosper.org/semihexesmedian.png> most resembles the Macsyma plot. Since Julian recently found a sampling of Heighway's Dragon equal to its median curve, it's likely that there are samplings of these semihexes coincident with their median curves. This will be easy to test when we manage to invert the semihex functions. Which will also let us directly exhibit the sextuple points. Failing that, it should be possible to tediously guess and check a few. (Actually, since the Dragon truly spacefills, there is obviously some way to hit all the vertices on the median curve. But "luckily" not crazy and disordered.)
Heretofore, the only spacefill dense with sextuple points I'd seen was Julian's huge Fourier triangle (@http://howwords.com/triangles/artcircle, which seems to be down), made from 25 copies of itself. Unfortunately, sextuple points are visited very furtively, so the Fourier plot is unconvincing, never (in six tries) coming very close to the actual sextuple points. For the exact definition, see my 21 Aug 2016 math-fun, which gives the six preimages of In[381]:= untrifil25[1/2 + I Sqrt[3]/10]
Out[381]= {8/75, 11/75, 14/75, 17/75, 4/15, 23/75} (In arithmetic progression. Always?) —rwg
That crab chain fractal is actually a closed loop—a fact concealed by dyadic rational sampling, which collapses small loops into retraced line segments. To see loops <http://gosper.org/crabsamps.png>, throw into the sampling frequency a factor of 3 or 5. Just don't tell your shrink what they remind you of. —rwg On Sun, Dec 1, 2019 at 4:45 PM Bill Gosper <billgosper@gmail.com> wrote:
Argh, I'm an idiot. False alarm: 1+i√3/2 is not a vertex of either semihex! It's in the east quadrant of the closed (upper) loop. Apologies to all who were contemplating a psychiatric intervention. Here are some views in one zoomable frame. <http://gosper.org/sextups.png> Note that three of the six grazing regions resemble daisy-chained horseshoe crabs. Do they look familiar to anyone? (Other than crab voyeurs,) —rwg
On Sat, Nov 30, 2019 at 10:07 PM Bill Gosper <billgosper@gmail.com> wrote:
I confess to being too stupid to construct the inversesemihexes function, even with Julian's magic toolbox. But, spazzing around, I found that a hexagon *vertex* is a sextuple point!?
In[483]:= Cases[Table[{t, semihexes@t}, {t, 0, 1, 1/64/15}], {_, {1 + I √3/2}}]
Out[483]=
{{21/80, {1 + (I √3)/2}},{43/160, {1 + (I √3)/2}}, {13/40, {1 + (I √3)/2}},
{53/160, {1 + (I √3)/2}}, {77/160, {1 + (I √3)/2}},{39/80, {1 + (I √3)/2}}}
This seems to say that if you extend the curve to make three copies
sharing that point, it would be 18-tuple!? I find this incredible.
—rwg
On Sat, Nov 30, 2019 at 12:20 AM Bill Gosper <billgosper@gmail.com> wrote:
Julian & I (again mostly Julian) found exactly the two continuous functions schematized by this old Macsyma plot <http://gosper.org/hemis.gif> of a closed and unclosed curve. So now we can sample the functions at arbitrary rationals. E.g., triple frequency <http://gosper.org/semihexes3x.png>. But actually the median curve <http://gosper.org/semihexesmedian.png> most resembles the Macsyma plot. Since Julian recently found a sampling of Heighway's Dragon equal to its median curve, it's likely that there are samplings of these semihexes coincident with their median curves. This will be easy to test when we manage to invert the semihex functions. Which will also let us directly exhibit the sextuple points. Failing that, it should be possible to tediously guess and check a few. (Actually, since the Dragon truly spacefills, there is obviously some way to hit all the vertices on the median curve. But "luckily" not crazy and disordered.)
Heretofore, the only spacefill dense with sextuple points I'd seen was Julian's huge Fourier triangle (@http://howwords.com/triangles/artcircle, which seems to be down), made from 25 copies of itself. Unfortunately, sextuple points are visited very furtively, so the Fourier plot is unconvincing, never (in six tries) coming very close to the actual sextuple points. For the exact definition, see my 21 Aug 2016 math-fun, which gives the six preimages of In[381]:= untrifil25[1/2 + I Sqrt[3]/10]
Out[381]= {8/75, 11/75, 14/75, 17/75, 4/15, 23/75} (In arithmetic progression. Always?) —rwg
Howard Cannon's Ptolemizer sandbox (e.g. http://howwords.com/triangles/?a_=sexttri&radiusPct=90&nRotors=3125&dtPct=20... ) is back up. Play with it! —rwg On Sun, Dec 1, 2019 at 8:39 PM Bill Gosper <billgosper@gmail.com> wrote:
That crab chain fractal is actually a closed loop—a fact concealed by dyadic rational sampling, which collapses small loops into retraced line segments. To see loops <http://gosper.org/crabsamps.png>, throw into the sampling frequency a factor of 3 or 5. Just don't tell your shrink what they remind you of. —rwg
On Sun, Dec 1, 2019 at 4:45 PM Bill Gosper <billgosper@gmail.com> wrote:
Argh, I'm an idiot. False alarm: 1+i√3/2 is not a vertex of either semihex! It's in the east quadrant of the closed (upper) loop. Apologies to all who were contemplating a psychiatric intervention. Here are some views in one zoomable frame. <http://gosper.org/sextups.png> Note that three of the six grazing regions resemble daisy-chained horseshoe crabs. Do they look familiar to anyone? (Other than crab voyeurs,) —rwg
On Sat, Nov 30, 2019 at 10:07 PM Bill Gosper <billgosper@gmail.com> wrote:
I confess to being too stupid to construct the inversesemihexes function, even with Julian's magic toolbox. But, spazzing around, I found that a hexagon *vertex* is a sextuple point!?
In[483]:= Cases[Table[{t, semihexes@t}, {t, 0, 1, 1/64/15}], {_, {1 + I √3/2}}]
Out[483]=
{{21/80, {1 + (I √3)/2}},{43/160, {1 + (I √3)/2}}, {13/40, {1 + (I √3)/2}},
{53/160, {1 + (I √3)/2}}, {77/160, {1 + (I √3)/2}},{39/80, {1 + (I √3)/2}}}
This seems to say that if you extend the curve to make three copies
sharing that point, it would be 18-tuple!? I find this incredible.
—rwg
On Sat, Nov 30, 2019 at 12:20 AM Bill Gosper <billgosper@gmail.com> wrote:
Julian & I (again mostly Julian) found exactly the two continuous functions schematized by this old Macsyma plot <http://gosper.org/hemis.gif> of a closed and unclosed curve. So now we can sample the functions at arbitrary rationals. E.g., triple frequency <http://gosper.org/semihexes3x.png>. But actually the median curve <http://gosper.org/semihexesmedian.png> most resembles the Macsyma plot. Since Julian recently found a sampling of Heighway's Dragon equal to its median curve, it's likely that there are samplings of these semihexes coincident with their median curves. This will be easy to test when we manage to invert the semihex functions. Which will also let us directly exhibit the sextuple points. Failing that, it should be possible to tediously guess and check a few. (Actually, since the Dragon truly spacefills, there is obviously some way to hit all the vertices on the median curve. But "luckily" not crazy and disordered.)
Heretofore, the only spacefill dense with sextuple points I'd seen was Julian's huge Fourier triangle (@http://howwords.com/triangles/artcircle, which seems to be down), made from 25 copies of itself. Unfortunately, sextuple points are visited very furtively, so the Fourier plot is unconvincing, never (in six tries) coming very close to the actual sextuple points. For the exact definition, see my 21 Aug 2016 math-fun, which gives the six preimages of In[381]:= untrifil25[1/2 + I Sqrt[3]/10]
Out[381]= {8/75, 11/75, 14/75, 17/75, 4/15, 23/75} (In arithmetic progression. Always?) —rwg
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