August 2016 - mathfun@mailman.xmission.com

[math-fun] Julian has constructed a spacefill dense with sextuple points
by Bill Gosper 26 Jan '17

26 Jan '17

using a 25-way conditional. ClearAll[piecewiserecursivefractal]; piecewiserecursivefractal[x_, f_, which_, iters_, fns_] := piecewiserecursivefractal[x, g_, which, iters, fns] = ((piecewiserecursivefractal[x, h_, which, iters, fns] := Block[{y}, y /. Solve[f[y] == h[y], y]]); Union @@ ((fns[[#]] /@piecewiserecursivefractal[iters[[#]][x], Composition[f, fns[[#]]], which, iters, fns]) & /@which[x])); (I cannot praise this function enough.) In[363]:= trifil25[t_] := piecewiserecursivefractal[t, Identity, Piecewise[{{{1}, #1 == 0}, {{Ceiling[25*#1]}, Inequality[0, Less, #1, LessEqual, 1]}}, {}] & , {25*#1 + 0 & , -25*#1 + 2 & , 25*#1 - 2 & , 25*#1 - 3 & , 25*#1 - 4 & , 25*#1 - 5 & , 25*#1 - 6 & , 25*#1 - 7 & , 25*#1 - 8 & , -25*#1 + 10 & , -25*#1 + 11 & , 25*#1 - 11 & , -25*#1 + 13 & , -25*#1 + 14 & , 25*#1 - 14 & , -25*#1 + 16 & , 25*#1 - 16 & , -25*#1 + 18 & , 25*#1 - 18 & , 25*#1 - 19 & , -25*#1 + 21 & , -25*#1 + 22 & , 25*#1 - 22 & , -25*#1 + 24 & , 25*#1 - 24 & }, {#1/5 & , (1/10)*(3 + I*Sqrt[3] + (-1 - I*Sqrt[3])* #1) & , (1/10)*(3 + I*Sqrt[3] + #1 - I*Sqrt[3]*#1) & , (2 + #1)/5 & , (1/10)*(6 + #1 + I*Sqrt[3]*#1) & , (1/10)*(7 + I*Sqrt[3] + I*(I + Sqrt[3])*#1) & , (1/5)*(3 + I*Sqrt[3] - #1) & , (1/10)*(4 + 2*I*Sqrt[3] + (-1 - I*Sqrt[3])*#1) & , (1/10)*(3 + I*Sqrt[3] - 2*#1) & , (-(1/10))*I*(-I + Sqrt[3])*(-2 + #1) & , (1/5)*(2 + I*Sqrt[3] - #1) & , (1/5)*(2 + I*Sqrt[3] + #1) & , (1/10)*(5 + 3*I*Sqrt[3] + #1 - I*Sqrt[3]*#1) & , (1/10)*(4 + 2*I*Sqrt[3] + #1 + I*Sqrt[3]*#1) & , (1/10)*(4 + 2*I*Sqrt[3] + I*(I + Sqrt[3])*#1) & , (-(1/10))*I*(-I + Sqrt[3])*(-4 + #1) & , (1/5)*(2 + 2*I*Sqrt[3] + #1) & , (1/10)*(5 + 3*I*Sqrt[3] + #1 + I*Sqrt[3]*#1) & , (1/10)*(5 + 3*I*Sqrt[3] + 2*#1) & , (1/10)*(7 + 3*I*Sqrt[3] + (-1 - I*Sqrt[3])*#1) & , (1/5)*(4 + I*Sqrt[3] - #1) & , (1/10)*(9 + I*Sqrt[3] + I*(I + Sqrt[3])*#1) & , (1/10)*(9 + I*Sqrt[3] - 2*#1) & , (1/10)*(8 + I*(I + Sqrt[3])*#1) & , (4 + #1)/5 & }][[1]] E.g., In[361]:= trifil25[2/25 + 2/75] // ComplexExpand Out[361]= {1/2 + (I Sqrt[3])/10} In[367]:= Graphics[{Hue@#, Line[ReIm@{trifil25@#, trifil25[# + 5^-5]}]} & /@ Range[0, 1 - 5^-5, 5^-5]] gosper.org/trifil25.png (Those little white hexagons are sextuple points.) The inverse fractal, constructed more or less automatically: In[380]:= untrifil25[t_] := piecewiserecursivefractal[t, Identity, If[0 <= Re[#1] <= 1 && 0 <= Im[#1] <= Sqrt[3]/2, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, {}] &, {ComplexExpand[5 #1] &, ComplexExpand[1/4 (6 - 2 I Sqrt[3] - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (-4 I Sqrt[3] + 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[-2 + 5 #1] &, ComplexExpand[1/4 (-6 + 6 I Sqrt[3] + 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (4 + 8 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[3 + I Sqrt[3] - 5 #1] &, ComplexExpand[1/4 (10 - 2 I Sqrt[3] - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[1/2 (3 + I Sqrt[3] - 10 #1)] &, ComplexExpand[1/4 (8 - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[2 + I Sqrt[3] - 5 #1] &, ComplexExpand[-2 - I Sqrt[3] + 5 #1] &, ComplexExpand[1/4 (4 - 8 I Sqrt[3] + 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (-10 + 2 I Sqrt[3] + 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (-2 + 6 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (16 - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[-2 - 2 I Sqrt[3] + 5 #1] &, ComplexExpand[1/4 (-14 + 2 I Sqrt[3] + 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/2 (-3 I Sqrt[3] + 5 (-1 + 2 #1))] &, ComplexExpand[1/4 (16 - 4 I Sqrt[3] - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[4 + I Sqrt[3] - 5 #1] &, ComplexExpand[1/4 (6 + 10 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/2 (9 + I Sqrt[3] - 10 #1)] &, ComplexExpand[1/4 (8 + 8 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[-4 + 5 #1] &}, {#1/25 &, (2 - #1)/25 &, (2 + #1)/ 25 &, (3 + #1)/25 &, (4 + #1)/25 &, (5 + #1)/25 &, (6 + #1)/ 25 &, (7 + #1)/25 &, (8 + #1)/25 &, (10 - #1)/25 &, (11 - #1)/ 25 &, (11 + #1)/25 &, (13 - #1)/25 &, (14 - #1)/25 &, (14 + #1)/ 25 &, (16 - #1)/25 &, (16 + #1)/25 &, (18 - #1)/25 &, (18 + #1)/ 25 &, (19 + #1)/25 &, (21 - #1)/25 &, (22 - #1)/25 &, (22 + #1)/ 25 &, (24 - #1)/25 &, (24 + #1)/25 &}] Here it finds the preimages of the simplest sextuple point: In[381]:= untrifil25[1/2 + I Sqrt[3]/10] Out[381]= {8/75, 11/75, 14/75, 17/75, 4/15, 23/75} --rwg

3 5

[math-fun] A problem about finite sets
by Victor Miller 10 Sep '16

10 Sep '16

Say that a finite set of integers, A, has *property P* is there are two non-empty disjoint subsets S and T of A, such that sum(S) = sum(T) (where sum(X) is the sum of the elements of X). 1) Show that every 10 element subset of {1,...,100} has property P (this is fairly easy). 2) Is it true that every 9 element subset of {1,...,100} has property P? Victor

8 9

[math-fun] Palo Alto to enforce zoning ban on software coding
by Henry Baker 01 Sep '16

01 Sep '16

FYI -- [Micro-aggression warning: undecidability lurks below.] Sheesh! Talk about heavy handed; Palo Alto could simply have bounded time and/or space for its Turing Machines. But NOoooh! they've prohibited *stacks* (no CF), and probably even finite state machines. I presume that this is another attack on encryption technology (it IS aimed at Palantir -- which occupies much of Palo Alto -- among other companies), but perhaps that is just paranoia. I believe that the zoning ban also applies to macros -- code-producing code -- and compilers. Perhaps it's time to resurrect Google's Ark -- Google's plan to station a floating high-tech development platform off-shore to get around H1B visa hiring restrictions. http://www.siliconbeat.com/2016/08/31/no-coding-palo-alto-city-takes-silico… No coding in Palo Alto? The birthplace of Hewlett Packard and Xerox Parc and founding place of Facebook is now considering whether to enforce a zoning regulation banning firms whose "primary business is research and development, including software coding," according to the New York Times. Palo Alto Mayor Patrick Burt told the Times: "Big tech companies are choking off the downtown. Its not healthy." "For many of the 66,000 residents of Palo Alto, all this coding and innovation may be a bit too much." "But Palo Alto may not want the jobs either. Making coding off-limits in certain sections, however that is enforced, sends that message." http://www.nytimes.com/2016/08/30/us/spotty-cell-reception-in-the-heart-of-… "the mayor [of Palo Alto] is looking to enforce, in some form, an all-but-forgotten zoning regulation that bans companies whose primary business is research and development, including software coding."

4 3

Re: [math-fun] Julian has constructed a spacefill dense with sextuple points
by Bill Gosper 31 Aug '16

31 Aug '16

Hi Vitaliy, I have a sandbox with a general Fourier animator. It takes a formula for the coefficients (amplitudes and phases as functions of frequency), speed, and coloration. It isn't very fast because it colors one pixel at a time, emulating the Symbolics mathematically correct draw-triangle ALU-add microcode necessary for smoothly drawing moving edges. Mathematically correct means that, if two triangles share an edge, there will never be missing or overwritten pixels, and if the triangle is "inside out", the "bump" will be negated. (The trick is to consistently define exactly which pixels to bump.) With this primitive, you can then animate the movement of an edge segment simply by drawing two triangles filling the quadrilateral defined by the old and new positions of its endpoints. To draw a seamless T joint, just draw the "zero area" triangle formed by the constituent edges, which will usually add or subtract the "bump" constant to or from a few pixels. --Bill On Wed, Aug 31, 2016 at 2:51 PM, Vitaliy Kaurov <vitaliyk(a)wolfram.com> wrote: > Bill do you have full Mathematica code for these animations? > > for the dyadic dragon (upper right element, /. a -> π/4 for true > spacefill) > gosper.org/DDrag.mp4 > nor the peculiar 2x2 in > gosper.org/gaskettalk.pdf > (slides) for the Sierpinski Gasket & Co.: > gosper.org/Base2alores-1.wmv > gosper.org/Base2hires.mp4 > (digits = {0,1,ω,ω^2}.) > > > Cheers, > Vitaliy > > [chop] >

1 0

[math-fun] Julian's sexy spacefiller
by Bill Gosper 27 Aug '16

27 Aug '16

maps [2/25,8/25] onto a regular hexagon: gosper.org/3125hex.png Here's a sampling with small dinosaurs surrounding a ghostly hexagon at an odd angle: gosper.org/ghosthex.png Here are some beat frequency effects: gosper.org/beatfil.png And a remarkably celebratory finale: gosper.org/partyfil.png --rwg

1 0

[math-fun] Bizarro arithmetic
by James Propp 25 Aug '16

25 Aug '16

Has anyone written in an accessible vein about all the bad things that happen when you decree that minus times minus equals minus instead of plus? I might do this in my September blog post but I'm hoping someone else has beaten me to it. Jim Propp

6 12

[math-fun] Sign conventions
by James Propp 25 Aug '16

25 Aug '16

What's the canonical example of two scientific communities that have adopted mutually inconsistent sign-conventions? (Do physicists and electrical engineers describe electric fields using different sign conventions, maybe?) This relates to my most recent "bizarro arithmetic" post inasmuch as Martinez seems to be inviting us to imagine that normal and bizarro arithmetic could be set up as separate self-consistent systems, with a dictionary that lets you convert true statements in one system into true statements in the other. I'll believe it when I see it... Jim Propp

3 3

Re: [math-fun] Julian has constructed a spacefill dense with sextuple points
by Bill Gosper 24 Aug '16

24 Aug '16

On 2016-08-24 04:32, James Propp wrote: > Not surprisingly, Bill's identities and pictures are beautiful. But unless > I skimmed overhastily, none of the pictures depict the Fourier transforms > themselves in a fashion that would give an immediate visual answer to my > question about anisotropy. > > I'm also not sure how Bill is defining the Fourier transform (due to lazy > skimming on my part, I'm sure). Is he integrating a delta-function > concentrated on the set, whose "dimensionality of delta-ness" is calibrated > to ensure that the relevant integrals neither vanish nor blow up? If not, > how is the transform defined? > > Jim Propp It uses a (possibly new) technique to express the Fourier integrals directly as infinite matrix products, which, for the Snowflake- and Triadic Dragon-like examples in the paper (which is rather unskimmable), telescope to ordinary infinite products. I have never succeeded in telescoping the 3x3 product _n Sec[a]/4*{{Exp[-I*a], -Exp[I*(a - t)], 2*I*Sin[a + t/2]*Exp[-I*t/2]}, {-Exp[I*a], Exp[-I*(a + t)], 2*I*Sin[a - t/2]*Exp[-I*t/2]}, {0, 0, 4*Cos[a]*Cos[t/2]*Exp[-I*t/2]}}/. t -> π x/2^n for the dyadic dragon (upper right element, /. a -> π/4 for true spacefill) gosper.org/DDrag.mp4 nor the peculiar 2x2 in gosper.org/gaskettalk.pdf (slides) for the Sierpinski Gasket & Co.: gosper.org/Base2alores-1.wmv gosper.org/Base2hires.mp4 (digits = {0,1,ω,ω^2}.) --rwg > > On Wed, Aug 24, 2016 at 6:28 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > >> On 2016-08-23 12:51, James Propp wrote: >> > Take the graph of a square-filling parametrized path from [0,1] onto >> > [0,1]x[0,1], as an object in [0,1]^3; does it in any meaningful sense >> have >> > a Fourier transform? >> >> Yes! gosper.org/fst.pdf >> gives the Fourier series for various dragons, snowflakes, and carpets. I'm >> pretty sure I used those techniques on the Hilbert, but I'll probably have >> to >> rederive it. >> >> >If so, perhaps the anisotropy of the graph would show >> > up in the transform in some more localized fashion. >> > >> > Jim Propp >> >> Thank you for thinking of this. I've been wondering what happens to the >> area >> enclosed by, say, the closed loop of Hilbert's "curve" between preimages >> 1/6 >> and 1/2 as the sampling frequency → ∞. >> gosper.org/prf.pdf >> For finite samplings and finite Fourier >> approximations, the enclosed area approaches half the area enclosed by the >> progression of quadrants, >> = ½ (¼.+ 1/16+ 1/64+ ...) = 1/6. But in the limit, the interior and >> exterior >> suddenly disappear, and everything becomes boundary. >> >> There is a formula for the area enclosed by a curve described by a Fourier >> series. It will be interesting to see if it converges, almost certainly to >> ⅙, >> as the sampling frequency → ∞. Infinite sums are by definition limits. >> They can't suddenly jump when they reach ∞. OtOH, Fourier approximations >> can be unfaithful. Is there some fancy measure theory, applied to the >> actual >> Hilbert function, that gets a different answer? Say, 0? >> > >> > On Tue, Aug 23, 2016 at 1:43 PM, Joerg Arndt <arndt(a)jjj.de> wrote: >> > >> >> * Bill Gosper <billgosper(a)gmail.com> [Aug 23. 2016 19:35]: >> >> > [...] >> >> > Technically, these functions _are_ curves, being continuous images of >> >> one- >> >> > dimensional sets. But people confuse curves with their graphs, and >> their >> >> > graphs are blobs. >> But ironically, nonblobs should be invisible. "If you can see a line >> segment, >> you've drawn it wrong." The only accurate picture I can think of of a >> connected 1D set is as the boundary of a 2D area. But what if it's an >> unclosed >> arc? For B&W, a nice solution might be to shade one side of the curve, >> fading >> from black to white with distance from the boundary. (Not always easily >> computed.) More stylish: Gray background. Fade from white to gray on >> one >> side; black to gray on the other. --rwg >> >> > [...] >> >> >> >> Isn't that a non-problem resolved by differentiating between the >> >> "finite approximations" (certainly curves!) and the limits? >> >> Note how I dance around that by saying "shapes" (limits) and >> >> "iterates" (curves) in my arXiv paper. >> >> >> >> Best regards, jj >> >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

1 0

Re: [math-fun] Julian has constructed a spacefill dense with sextuple points
by Bill Gosper 24 Aug '16

24 Aug '16

On 2016-08-23 12:51, James Propp wrote: > Take the graph of a square-filling parametrized path from [0,1] onto > [0,1]x[0,1], as an object in [0,1]^3; does it in any meaningful sense have > a Fourier transform? Yes! gosper.org/fst.pdf gives the Fourier series for various dragons, snowflakes, and carpets. I'm pretty sure I used those techniques on the Hilbert, but I'll probably have to rederive it. >If so, perhaps the anisotropy of the graph would show > up in the transform in some more localized fashion. > > Jim Propp Thank you for thinking of this. I've been wondering what happens to the area enclosed by, say, the closed loop of Hilbert's "curve" between preimages 1/6 and 1/2 as the sampling frequency → ∞. gosper.org/prf.pdf For finite samplings and finite Fourier approximations, the enclosed area approaches half the area enclosed by the progression of quadrants, = ½ (¼.+ 1/16+ 1/64+ ...) = 1/6. But in the limit, the interior and exterior suddenly disappear, and everything becomes boundary. There is a formula for the area enclosed by a curve described by a Fourier series. It will be interesting to see if it converges, almost certainly to ⅙, as the sampling frequency → ∞. Infinite sums are by definition limits. They can't suddenly jump when they reach ∞. OtOH, Fourier approximations can be unfaithful. Is there some fancy measure theory, applied to the actual Hilbert function, that gets a different answer? Say, 0? > > On Tue, Aug 23, 2016 at 1:43 PM, Joerg Arndt <arndt(a)jjj.de> wrote: > >> * Bill Gosper <billgosper(a)gmail.com> [Aug 23. 2016 19:35]: >> > [...] >> > Technically, these functions _are_ curves, being continuous images of >> one- >> > dimensional sets. But people confuse curves with their graphs, and their >> > graphs are blobs. But ironically, nonblobs should be invisible. "If you can see a line segment, you've drawn it wrong." The only accurate picture I can think of of a connected 1D set is as the boundary of a 2D area. But what if it's an unclosed arc? For B&W, a nice solution might be to shade one side of the curve, fading from black to white with distance from the boundary. (Not always easily computed.) More stylish: Gray background. Fade from white to gray on one side; black to gray on the other. --rwg >> > [...] >> >> Isn't that a non-problem resolved by differentiating between the >> "finite approximations" (certainly curves!) and the limits? >> Note how I dance around that by saying "shapes" (limits) and >> "iterates" (curves) in my arXiv paper. >> >> Best regards, jj >>

2 1

Re: [math-fun] Julian has constructed a spacefill dense with sextuple points
by Bill Gosper 23 Aug '16

23 Aug '16

On 2016-08-23 04:33, James Propp wrote: > I uagree with everything Bill says about space-filling curves not being > honest curves, but I don't understand what he means by "No" (his first > word). Did I get something wrong in my most recent email? > > Still wondering wondering what part of "No" I didn't understand, > > Jim Propp Well, maybe I misunderstood "fill all of a discrete plane" to mean some kind of (non dense) grid of separate points, while you meant a topology where all sets are open. The former view produces lots of eye-catching but misleading pictures. L-systems are just schematics. Can they really define functions R -> C by taking limits? How could you use them to verify that, say, the only solutions of Hilbert(x) = x+i are the Thue-Morse constant and its one's complement? --rwg Technically, these functions _are_ curves, being continuous images of one- dimensional sets. But people confuse curves with their graphs, and their graphs are blobs. > > On Tue, Aug 23, 2016 at 5:24 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > >> No. >> This is why I hate calling these functions "curves". They >> map closed intervals onto closed areas of C. This is why they >> are dense with double and triple points. --rwg >> >> On 2016-08-22 16:20, James Propp wrote: >> > I misunderstood what game you all were playing, so my question was a bit >> > off base. >> > >> > Joerg and Bill et al. are looking at ways to fill all of a discrete >> plane; >> > I was thinking more about Peano curves and the like, which fill a finite >> > portion of a continuous plane. >> > >> > Jim >> > >> > On Monday, August 22, 2016, James Propp <jamespropp(a)gmail.com> wrote: >> > >> >> Mulling over what it might mean to say that a spacefilling curve lives >> on >> >> a grid, I came up with a nonstandard topology to put on R^2. >> >> >> >> The basic open sets are open line segments. This is a finer topology >> than >> >> the standard one. What mappings from [0,1] into R^2 are continuous in >> this >> >> topology? Piecewise-polygonal ones; what else? Does this topology have >> any >> >> uses? >> >> >> >> But this doesn't address the question that started me down this line of >> >> thought, namely: what might it mean to say that a spacefilling curve >> lives >> >> on a grid, or prefers certain directions? The approximating polygonal >> >> curves may use only certain directions, but how does this property >> manifest >> >> itself with regard to the limit-object? >> >> >> >> Jim Propp >> >> >> >> On Mon, Aug 22, 2016 at 3:26 AM, Joerg Arndt <arndt(a)jjj.de >> >> <javascript:_e(%7B%7D,'cvml','arndt(a)jjj.de');>> wrote: >> >> >> >>> For similar curves (that exist for orders that are odd squares) >> >>> on the (3.6.3.6)-grid see >> >>> http://jjj.de/3frac/p6/all-r25-b-curve-decompositions.pdf (last two >> >>> pages) >> >>> and >> >>> http://jjj.de/3frac/p6/all-r25-b-tiles.pdf (last four pages). >> >>> >> >>> Sextuple and triple points are easy to see in these. >> >>> >> >>> I cannot decode the Mathca, does you curve live >> >>> on the (3.6.3.6)-grid as well (morally it should)? >> >>> >> >>> Best regards, jj >> >>> >> >>> >> >>> * Bill Gosper <billgosper(a)gmail.com >> >>> <javascript:_e(%7B%7D,'cvml','billgosper(a)gmail.com');>> [Aug 22. 2016 >> >>> 08:17]: >> >>> > using a 25-way conditional. >> >>> > ClearAll[piecewiserecursivefractal]; >> >>> > piecewiserecursivefractal[x_, f_, which_, iters_, fns_] := >> >>> > piecewiserecursivefractal[x, g_, which, iters, >> >>> > fns] = ((piecewiserecursivefractal[x, h_, which, iters, fns] := >> >>> > Block[{y}, y /. Solve[f[y] == h[y], y]]); >> >>> > Union @@ ((fns[[#]] /@piecewiserecursivefractal[iters[[#]][x], >> >>> > Composition[f, fns[[#]]], which, iters, fns]) & >> /@which[x])); >> >>> > >> >>> > (I cannot praise this function enough.) >> >>> > >> >>> > In[363]:= trifil25[t_] := piecewiserecursivefractal[t, Identity, >> >>> > Piecewise[{{{1}, #1 == 0}, {{Ceiling[25*#1]}, >> >>> > Inequality[0, Less, #1, LessEqual, 1]}}, {}] & , >> >>> > {25*#1 + 0 & , -25*#1 + 2 & , 25*#1 - 2 & , 25*#1 - 3 & , >> >>> > 25*#1 - 4 & , 25*#1 - 5 & , 25*#1 - 6 & , 25*#1 - 7 & , >> >>> > 25*#1 - 8 & , -25*#1 + 10 & , -25*#1 + 11 & , >> >>> > 25*#1 - 11 & , -25*#1 + 13 & , -25*#1 + 14 & , >> >>> > 25*#1 - 14 & , -25*#1 + 16 & , 25*#1 - 16 & , >> >>> > -25*#1 + 18 & , 25*#1 - 18 & , 25*#1 - 19 & , >> >>> > -25*#1 + 21 & , -25*#1 + 22 & , 25*#1 - 22 & , >> >>> > -25*#1 + 24 & , 25*#1 - 24 & }, >> >>> > {#1/5 & , (1/10)*(3 + I*Sqrt[3] + (-1 - I*Sqrt[3])* >> >>> > #1) & , (1/10)*(3 + I*Sqrt[3] + #1 - >> >>> > I*Sqrt[3]*#1) & , (2 + #1)/5 & , >> >>> > (1/10)*(6 + #1 + I*Sqrt[3]*#1) & , >> >>> > (1/10)*(7 + I*Sqrt[3] + I*(I + Sqrt[3])*#1) & , >> >>> > (1/5)*(3 + I*Sqrt[3] - #1) & , >> >>> > (1/10)*(4 + 2*I*Sqrt[3] + (-1 - I*Sqrt[3])*#1) & , >> >>> > (1/10)*(3 + I*Sqrt[3] - 2*#1) & , >> >>> > (-(1/10))*I*(-I + Sqrt[3])*(-2 + #1) & , >> >>> > (1/5)*(2 + I*Sqrt[3] - #1) & , >> >>> > (1/5)*(2 + I*Sqrt[3] + #1) & , >> >>> > (1/10)*(5 + 3*I*Sqrt[3] + #1 - I*Sqrt[3]*#1) & , >> >>> > (1/10)*(4 + 2*I*Sqrt[3] + #1 + I*Sqrt[3]*#1) & , >> >>> > (1/10)*(4 + 2*I*Sqrt[3] + I*(I + Sqrt[3])*#1) & , >> >>> > (-(1/10))*I*(-I + Sqrt[3])*(-4 + #1) & , >> >>> > (1/5)*(2 + 2*I*Sqrt[3] + #1) & , >> >>> > (1/10)*(5 + 3*I*Sqrt[3] + #1 + I*Sqrt[3]*#1) & , >> >>> > (1/10)*(5 + 3*I*Sqrt[3] + 2*#1) & , >> >>> > (1/10)*(7 + 3*I*Sqrt[3] + (-1 - I*Sqrt[3])*#1) & , >> >>> > (1/5)*(4 + I*Sqrt[3] - #1) & , >> >>> > (1/10)*(9 + I*Sqrt[3] + I*(I + Sqrt[3])*#1) & , >> >>> > (1/10)*(9 + I*Sqrt[3] - 2*#1) & , >> >>> > (1/10)*(8 + I*(I + Sqrt[3])*#1) & , (4 + #1)/5 & }][[1]] >> >>> > E.g., >> >>> > In[361]:= trifil25[2/25 + 2/75] // ComplexExpand >> >>> > >> >>> > Out[361]= {1/2 + (I Sqrt[3])/10} >> >>> > >> >>> > In[367]:= Graphics[{Hue@#, >> >>> > Line[ReIm@{trifil25@#, trifil25[# + 5^-5]}]} & /@ Range[0, 1 - >> >>> 5^-5, >> >>> > 5^-5]] >> >>> > gosper.org/trifil25.png >> >>> > (Those little white hexagons are sextuple points.) >> >>> > >> >>> > The inverse fractal, constructed more or less automatically: >> >>> > In[380]:= untrifil25[t_] := >> >>> > piecewiserecursivefractal[t, Identity, >> >>> > If[0 <= Re[#1] <= 1 && 0 <= Im[#1] <= Sqrt[3]/2, {1, 2, 3, 4, 5, 6, >> >>> > 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, >> >>> > 24, 25}, {}] &, {ComplexExpand[5 #1] &, >> >>> > ComplexExpand[1/4 (6 - 2 I Sqrt[3] - 10 #1 + 10 I Sqrt[3] #1)] &, >> >>> > ComplexExpand[1/4 (-4 I Sqrt[3] + 10 #1 + 10 I Sqrt[3] #1)] &, >> >>> > ComplexExpand[-2 + 5 #1] &, >> >>> > ComplexExpand[1/4 (-6 + 6 I Sqrt[3] + 10 #1 - 10 I Sqrt[3] #1)] &, >> >>> > ComplexExpand[1/4 (4 + 8 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, >> >>> > ComplexExpand[3 + I Sqrt[3] - 5 #1] &, >> >>> > ComplexExpand[1/4 (10 - 2 I Sqrt[3] - 10 #1 + 10 I Sqrt[3] #1)] &, >> >>> > ComplexExpand[1/2 (3 + I Sqrt[3] - 10 #1)] &, >> >>> > ComplexExpand[1/4 (8 - 10 #1 + 10 I Sqrt[3] #1)] &, >> >>> > ComplexExpand[2 + I Sqrt[3] - 5 #1] &, >> >>> > ComplexExpand[-2 - I Sqrt[3] + 5 #1] &, >> >>> > ComplexExpand[1/4 (4 - 8 I Sqrt[3] + 10 #1 + 10 I Sqrt[3] #1)] &, >> >>> > ComplexExpand[1/4 (-10 + 2 I Sqrt[3] + 10 #1 - 10 I Sqrt[3] #1)] >> &, >> >>> > ComplexExpand[1/4 (-2 + 6 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] >> &, >> >>> > ComplexExpand[1/4 (16 - 10 #1 + 10 I Sqrt[3] #1)] &, >> >>> > ComplexExpand[-2 - 2 I Sqrt[3] + 5 #1] &, >> >>> > ComplexExpand[1/4 (-14 + 2 I Sqrt[3] + 10 #1 - 10 I Sqrt[3] #1)] >> &, >> >>> > ComplexExpand[1/2 (-3 I Sqrt[3] + 5 (-1 + 2 #1))] &, >> >>> > ComplexExpand[1/4 (16 - 4 I Sqrt[3] - 10 #1 + 10 I Sqrt[3] #1)] &, >> >>> > ComplexExpand[4 + I Sqrt[3] - 5 #1] &, >> >>> > ComplexExpand[1/4 (6 + 10 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, >> >>> > ComplexExpand[1/2 (9 + I Sqrt[3] - 10 #1)] &, >> >>> > ComplexExpand[1/4 (8 + 8 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, >> >>> > ComplexExpand[-4 + 5 #1] &}, {#1/25 &, (2 - #1)/25 &, (2 + #1)/ >> >>> > 25 &, (3 + #1)/25 &, (4 + #1)/25 &, (5 + #1)/25 &, (6 + #1)/ >> >>> > 25 &, (7 + #1)/25 &, (8 + #1)/25 &, (10 - #1)/25 &, (11 - #1)/ >> >>> > 25 &, (11 + #1)/25 &, (13 - #1)/25 &, (14 - #1)/25 &, (14 + #1)/ >> >>> > 25 &, (16 - #1)/25 &, (16 + #1)/25 &, (18 - #1)/25 &, (18 + #1)/ >> >>> > 25 &, (19 + #1)/25 &, (21 - #1)/25 &, (22 - #1)/25 &, (22 + #1)/ >> >>> > 25 &, (24 - #1)/25 &, (24 + #1)/25 &}] >> >>> > >> >>> > Here it finds the preimages of the simplest sextuple point: >> >>> > >> >>> > In[381]:= untrifil25[1/2 + I Sqrt[3]/10] >> >>> > >> >>> > Out[381]= {8/75, 11/75, 14/75, 17/75, 4/15, 23/75} >> >>> > --rwg

3 2