Re: [math-fun] Fractal puzzles
Thanks, Alan.
So, does this make it the same as starting with a plus sign of 5 adjacent squares, and taking 5 congruent copies snuggled together and normalized -- then iterate to the limit? (Actually the boundary of this limit.)
If you do it right, you can start with a square for Q[0] and get the "+" sign for Q[1]. See http://gosper.org/erezd.PNG . These aren't quite the same but they have the same closure and boundary dimension. A toy analogy is the approximations to the unit interval you get by varying Q[0] in the iteration Q[n+1] = (Q[n] U (1+Q[n]))/2. Allan's method, Q[0] := {0}, produces the dyadic rationals. But Q[0] := (0,1] gives the complement of the dyadic rationals. Same closure, utterly unequal measures. You can't zoom down to a self contact because of self similarity. For greater rigor, you can construct sausage links that will remain demonstrably disjoint. You can also write a function analogous to that obscure Peano Mathematica function that will exactly and continuously map the rationals in the unit interval onto one quarter of the boundary in question. There are also open and closed loop spacefills of this region, which can also be computed exactly for rationals. I like D=log_5(9) . Any takers on 3D revisitation? --rwg
If so, it's a fractal I was studying just a couple of months ago. I'll go review what I had about it.
=-Dan
<< I'm pretty sure I understand the intended construction. We'll construct a sequence of sets of complex numbers; the limit of this sequence will be (well-defined and) the intended fractal.
Let Q[0] contain only 0. Then for any nonnegative integer i, let Q[i+1] = (Q[i] + {0, 1, -1, i, -i}) / (2i + 1). Here, if A and B are sets, the set A+B is intended to mean {a+b | a in A and b in B}.
Each Q is roughly cross-shaped; RWG observes (very tersely) that dividing by 2i+1 rotates and shrinks each such cross by just enough that five crosses can snuggle together to make a meta-cross.
I think this fractal is in Mandelbrot; I cannot dig up my copy at the moment.
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I just got SUM(N/(%E^(2*SQRT(11)*%PI*N)-1),N,1,INF) = -SQRT(22^(1/3)*(2733*SQRT(33)+624899)^(1/3)+22^(1/3)*(624899-2733*SQRT(33))^(1/3)+240)*GAMMA(1/22)*GAMMA(3/22)*GAMMA(5/22)*GAMMA(9/22)*GAMMA(15/22)/(8448*SQRT(11)*%PI^(7/2))-1/(8*SQRT(11)*%PI)+1/24 inf ==== \ n 1/3
---------------------- = - sqrt(22 / 2 sqrt(11) %pi n ==== %e - 1 n = 1
1/3 1/3 1/3 (2733 sqrt(33) + 624899) + 22 (624899 - 2733 sqrt(33)) 1 3 5 9 15 + 240) gamma(--) gamma(--) gamma(--) gamma(--) gamma(--) 22 22 22 22 22 7/2 1 1 /(8448 sqrt(11) %pi ) - -------------- + -- 8 sqrt(11) %pi 24 and closed forms for eta(exp(-2 sqrt(11)pi)) and eta(exp(-2 sqrt(19)pi)). The logderivative sum needs another hour's work. Working toward sqrt(163)pi. --rwg
I just got SUM(N/(%E^(2*SQRT(11)*%PI*N)-1),N,1,INF) = -SQRT(22^(1/3)*(2733*SQRT(33)+624899)^(1/3)+22^(1/3)*(624899-2733*SQRT(33))^(1/3)+240)*GAMMA(1/22)*GAMMA(3/22)*GAMMA(5/22)*GAMMA(9/22)*GAMMA(15/22)/(8448*SQRT(11)*%PI^(7/2))-1/(8*SQRT(11)*%PI)+1/24
inf ==== \ n 1/3
---------------------- = - sqrt(22 / 2 sqrt(11) %pi n ==== %e - 1 n = 1
1/3 1/3 1/3 (2733 sqrt(33) + 624899) + 22 (624899 - 2733 sqrt(33))
1 3 5 9 15 + 240) gamma(--) gamma(--) gamma(--) gamma(--) gamma(--) 22 22 22 22 22
7/2 1 1 /(8448 sqrt(11) %pi ) - -------------- + -- 8 sqrt(11) %pi 24
and closed forms for eta(exp(-2 sqrt(11)pi)) and eta(exp(-2 sqrt(19)pi)). The logderivative sum needs another hour's work.
Eye mercy: http://gosper.org/newetas.html
Working toward sqrt(163)pi.
These are purely empirical, unproven results. Incredibly, I'm doing the numerics in Macsyma instead of Mma due to bizarre precision bugs. And bizarreness in general. Floor[<numeric infinite series>] gave no integer. N[%] does, but then N[%] again makes a short float! But Mma's algebraic number stuff is pretty impressive. Still doesn't denest, tho. I shouldn't jinx myself, but I think I can do exp(pi sqrt(n/d)) for n and d "within reason". If 163 is beyond reason, wait 'til next year. --rwg
* rwg@sdf.lonestar.org <rwg@sdf.lonestar.org> [Jul 02. 2009 09:56]:
[...]
Eye mercy: http://gosper.org/newetas.html
Working toward sqrt(163)pi.
These are purely empirical, unproven results. Incredibly, I'm doing the numerics in Macsyma instead of Mma due to bizarre precision bugs. And bizarreness in general. Floor[<numeric infinite series>] gave no integer. N[%] does, but then N[%] again makes a short float!
But Mma's algebraic number stuff is pretty impressive. Still doesn't denest, tho.
I shouldn't jinx myself, but I think I can do exp(pi sqrt(n/d)) for n and d "within reason". If 163 is beyond reason, wait 'til next year. --rwg
The following refs might be helpful (wanted to work on that myself, don't have time). {Jinhee Yi: {Theta-function identities and the explicit formulas for theta-function and their applications}, Journal of Mathematical Analysis and Applications, vol.292, no.2, pp.381-400, \bdate{15-April-2004}. \jjfile{yi-theta-func-identities.pdf} % Seems to recycle vasuki-note-on-PQ-modeq.pdf {K.\ R.\ Vasuki, T.\ G.\ Sreeramamurthy: {A Note on $P$-$Q$ Modular Equations}, Tamsui Oxford Journal of Mathematical Sciences, vol.21, no.2, pp.109-120, \bdate{2005}. URL: \url{http://www.mcs.au.edu.tw/vol-21-2.htm}.} \jjfile{vasuki-note-on-PQ-modeq.pdf} % Much of this seems to be recycled in yi-theta-func-identities.pdf {M.\ S.\ Mahadeva, H.\ S.\ Madhusudhan : {Some explicit values for ratios of theta-functions}, General Mathematics, vol.13, no.2, pp.105-116, \bdate{2005}. URL: \url{http://www.emis.de/journals/GM/vol13nr2/cuprins132.html}.} \jjfile{mahadeva-some-explicit-theta-values.pdf}
* rwg@sdf.lonestar.org <rwg@sdf.lonestar.org> [Jul 02. 2009 09:56]:
[...]
Eye mercy: http://gosper.org/newetas.html
Working toward sqrt(163)pi.
These are purely empirical, unproven results. Incredibly, I'm doing the numerics in Macsyma instead of Mma due to bizarre precision bugs. And bizarreness in general. Floor[<numeric infinite series>] gave no integer. N[%] does, but then N[%] again makes a short float!
But Mma's algebraic number stuff is pretty impressive. Still doesn't denest, tho.
I shouldn't jinx myself, but I think I can do exp(pi sqrt(n/d)) for n and d "within reason". If 163 is beyond reason, wait 'til next year. --rwg
The following refs might be helpful (wanted to work on that myself, don't have time).
This first one looks most relevant. Is in on your site? Can you remind me the URL? --Bill
{Jinhee Yi: {Theta-function identities and the explicit formulas for theta-function and their applications}, Journal of Mathematical Analysis and Applications, vol.292, no.2, pp.381-400, \bdate{15-April-2004}. \jjfile{yi-theta-func-identities.pdf} % Seems to recycle vasuki-note-on-PQ-modeq.pdf
{K.\ R.\ Vasuki, T.\ G.\ Sreeramamurthy: {A Note on $P$-$Q$ Modular Equations}, Tamsui Oxford Journal of Mathematical Sciences, vol.21, no.2, pp.109-120, \bdate{2005}. URL: \url{http://www.mcs.au.edu.tw/vol-21-2.htm}.} \jjfile{vasuki-note-on-PQ-modeq.pdf} % Much of this seems to be recycled in yi-theta-func-identities.pdf
{M.\ S.\ Mahadeva, H.\ S.\ Madhusudhan : {Some explicit values for ratios of theta-functions}, General Mathematics, vol.13, no.2, pp.105-116, \bdate{2005}. URL: \url{http://www.emis.de/journals/GM/vol13nr2/cuprins132.html}.} \jjfile{mahadeva-some-explicit-theta-values.pdf}
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* rwg@sdf.lonestar.org <rwg@sdf.lonestar.org> [Jul 02. 2009 14:56]:
* rwg@sdf.lonestar.org <rwg@sdf.lonestar.org> [Jul 02. 2009 09:56]:
[...]
The following refs might be helpful (wanted to work on that myself, don't have time).
This first one looks most relevant. Is in on your site? Can you remind me the URL? --Bill
No open access it seems. Pay/subscriber URL is http://linkinghub.elsevier.com/retrieve/pii/S0022247X03009090 Sent paper in personal mail.
{Jinhee Yi: {Theta-function identities and the explicit formulas for theta-function and their applications}, Journal of Mathematical Analysis and Applications, vol.292, no.2, pp.381-400, \bdate{15-April-2004}. \jjfile{yi-theta-func-identities.pdf} % Seems to recycle vasuki-note-on-PQ-modeq.pdf
[...]
I just got SUM(N/(%E^(2*SQRT(11)*%PI*N)-1),N,1,INF) = -SQRT(22^(1/3)*(2733*SQRT(33)+624899)^(1/3)+22^(1/3)*(624899-2733*SQRT(33))^(1/3)+240)*GAMMA(1/22)*GAMMA(3/22)*GAMMA(5/22)*GAMMA(9/22)*GAMMA(15/22)/(8448*SQRT(11)*%PI^(7/2))-1/(8*SQRT(11)*%PI)+1/24
Joerg Arndt has just sent me a 2004 paper (THANKS!) by J. Yi actually deriving closed forms for theta[3](0,q) for q = +-exp(-n pi) for n as large as 12, in terms of <algebraic>*pi^(1/4)/Gamma(3/4). This is my first inkling that this eta stuff I've been bombarding you with isn't new. (Eavesdropper Bruce Berndt submitted the paper. Bruce, can you tell me what's known about Theta/Eta special values? (Not quotients.)) I got eta(exp(-pi Sqrt(43))) as a nice algebraic times a *horrible* pile of Gamma(k/86)^(n/10), for nearly all 0<k<43. The logs of all those Gammas resist numerical relation-finding. Those tenth roots are news, too. The Lambert series can't be much better: different algebraic times same Gammas to different powers. --rwg
Eye mercy: http://gosper.org/newetas.html
Does anybody really want to see eta(exp(-pi Sqrt(43)))? If you have Mma, DedekindEta[I*Sqrt[43]] == ((-(80/(1 + 63*Sqrt[129])^(1/3)) + (1 + 63*Sqrt[129])^(1/3))^(1/8)* Gamma[7/43]^(1/5)*Gamma[11/43]^(9/10)*Gamma[15/43]^(7/10)* (Gamma[4/43]*Gamma[6/43]*Gamma[10/43]*Gamma[14/43]* Gamma[16/43])^ (2/5)*(Gamma[3/43]*Gamma[19/43])^(3/5)* ((Gamma[1/86]*Gamma[9/86]*Gamma[13/86]*Gamma[17/86]* Gamma[21/86]* Gamma[25/86]*Gamma[41/86])/(Gamma[2/43]*Gamma[8/43]* Gamma[12/43]* Gamma[18/43]*Gamma[20/43]))^(1/10)* ((Gamma[1/43]*Gamma[5/43]*Gamma[9/43]*Gamma[13/43]* Gamma[17/43]* Gamma[21/43])/(Gamma[7/86]*Gamma[15/86]*Gamma[23/86]* Gamma[31/86]*Gamma[39/86]))^(3/10))/(2*86^(7/40)* Pi^(17/20)* Sqrt[Gamma[11/86]]*(Gamma[3/86]*Gamma[19/86])^(7/10)* Gamma[27/86]^(3/5)*Gamma[35/86]^(1/5)* (Gamma[5/86]*Gamma[29/86]*Gamma[33/86]*Gamma[37/86])^(2/5))
Working toward sqrt(163)pi.
It's threatening to be hideous. --rwg
These are purely empirical, unproven results. Incredibly, I'm doing the numerics in Macsyma instead of Mma due to bizarre precision bugs. And bizarreness in general. Floor[<numeric infinite series>] gave no integer. N[%] does, but then N[%] again makes a short float!
But Mma's algebraic number stuff is pretty impressive. Still doesn't denest, tho.
I shouldn't jinx myself, but I think I can do exp(pi sqrt(n/d)) for n and d "within reason". If 163 is beyond reason, wait 'til next year. --rwg
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I just got SUM(N/(%E^(2*SQRT(11)*%PI*N)-1),N,1,INF) = -SQRT(22^(1/3)*(2733*SQRT(33)+624899)^(1/3)+22^(1/3)*(624899-2733*SQRT(33))^(1/3)+240)*GAMMA(1/22)*GAMMA(3/22)*GAMMA(5/22)*GAMMA(9/22)*GAMMA(15/22)/(8448*SQRT(11)*%PI^(7/2))-1/(8*SQRT(11)*%PI)+1/24
Joerg Arndt has just sent me a 2004 paper (THANKS!) by J. Yi actually deriving closed forms for theta[3](0,q) for q = +-exp(-n pi) for n as large as 12, in terms of <algebraic>*pi^(1/4)/Gamma(3/4). This is my first inkling that this eta stuff I've been bombarding you with isn't new. (Eavesdropper Bruce Berndt submitted the paper. Bruce, can you tell me what's known about Theta/Eta special values? (Not quotients.))
I got eta(exp(-pi Sqrt(43))) as a nice algebraic times a *horrible* pile of Gamma(k/86)^(n/10), for nearly all 0<k<43. The logs of all those Gammas resist numerical relation-finding. Those tenth roots are news, too.
*Were* news. I got it about half as long with the empirical identity 2^(21/43) Pi^(3/2) Gamma[1/86] Gamma[3/86]^3 Gamma[5/86] Gamma[7/86]^2 Gamma[ 9/86] Gamma[11/86]^5 Gamma[13/86] Gamma[15/86]^2 Gamma[ 17/86] Gamma[19/86]^3 Gamma[21/86] Gamma[23/86]^2 Gamma[ 25/86] Gamma[27/86]^4 Gamma[29/86] Gamma[31/86]^2 Gamma[ 33/86] Gamma[35/86]^3 Gamma[37/86] Gamma[39/86]^2 Gamma[41/86] == Sqrt[43] Gamma[1/43]^2 Gamma[2/43] Gamma[3/43]^4 Gamma[ 4/43] Gamma[5/43]^2 Gamma[6/43] Gamma[7/43]^3 Gamma[ 8/43] Gamma[9/43]^2 Gamma[10/43] Gamma[11/43]^6 Gamma[ 12/43] Gamma[13/43]^2 Gamma[14/43] Gamma[15/43]^3 Gamma[ 16/43] Gamma[17/43]^2 Gamma[18/43] Gamma[19/43]^4 Gamma[ 20/43] Gamma[21/43]^2 This is presumably a nonobvious consequence of the ntuplication formula, presumably involving intermediate swell, since FullSimplify can't find it. There's probably a 43 <- n generalization that would be worth finding. E.g., 2^(20/41) Sqrt[41] Gamma[1/41]^4 Gamma[2/41] Gamma[3/41]^2 Gamma[4/41] Gamma[5/ 41]^3 Gamma[6/41] Gamma[7/41]^2 Gamma[8/41] Gamma[9/41]^6 Gamma[10/ 41] Gamma[11/41]^2 Gamma[12/41] Gamma[13/41]^3 Gamma[14/41] Gamma[ 15/41]^2 Gamma[16/41] Gamma[17/41]^4 Gamma[18/41] Gamma[19/ 41]^2 Gamma[20/41] == Pi Gamma[1/82]^3 Gamma[3/82] Gamma[5/ 82]^2 Gamma[7/82] Gamma[9/82]^5 Gamma[11/82] Gamma[13/82]^2 Gamma[ 15/82] Gamma[17/82]^3 Gamma[19/82] Gamma[21/82]^2 Gamma[23/ 82] Gamma[25/82]^4 Gamma[27/82] Gamma[29/82]^2 Gamma[31/82] Gamma[ 33/82]^3 Gamma[35/82] Gamma[37/82]^2 Gamma[39/82] But there are lots of these if we repeatedly drop the smallest arg and adjoin a new largest: (Sqrt[41] Gamma[1/41] Gamma[2/41] Gamma[3/41]^2 Gamma[4/41] Gamma[5/ 41]^3 Gamma[6/41] Gamma[7/41]^2 Gamma[8/41] Gamma[9/41]^6 Gamma[ 10/41] Gamma[11/41]^2 Gamma[12/41] Gamma[13/41]^3 Gamma[14/ 41] Gamma[15/41]^2 Gamma[16/41] Gamma[17/41]^4 Gamma[18/41] Gamma[ 19/41]^2 Gamma[20/41] Gamma[21/41]^3)== (4 2^(18/41) Pi^(5/2) Gamma[3/82] Gamma[5/82]^2 Gamma[7/82] Gamma[9/82]^5 Gamma[11/ 82] Gamma[13/82]^2 Gamma[15/82] Gamma[17/82]^3 Gamma[19/82] Gamma[ 21/82]^2 Gamma[23/82] Gamma[25/82]^4 Gamma[27/82] Gamma[29/ 82]^2 Gamma[31/82] Gamma[33/82]^3 Gamma[35/82] Gamma[37/ 82]^2 Gamma[39/82]) So it seems like any set of half the fractions in (0,n)/n will form a "basis", and by judiciously introducing fractions >1/2, it is likely possible to significantly reduce the number of different Gamma species. One of those n/2 choose n ought to be nice. Testing the shorter eta identity: In[117]:= {N[%], N[%, 33]} Out[117]= {False, True} Right helpful those Booleanizations. They drive my solving to Macsyma. --rwg
The Lambert series can't be much better: different algebraic times same Gammas to different powers.
Not so bad, now.
--rwg
Eye mercy: http://gosper.org/newetas.html
Updated.
Does anybody really want to see eta(exp(-pi Sqrt(43)))? If you have Mma, DedekindEta[I*Sqrt[43]] == ((-(80/(1 + 63*Sqrt[129])^(1/3)) + (1 + 63*Sqrt[129])^(1/3))^(1/8)* Gamma[7/43]^(1/5)*Gamma[11/43]^(9/10)*Gamma[15/43]^(7/10)* (Gamma[4/43]*Gamma[6/43]*Gamma[10/43]*Gamma[14/43]* Gamma[16/43])^ (2/5)*(Gamma[3/43]*Gamma[19/43])^(3/5)* ((Gamma[1/86]*Gamma[9/86]*Gamma[13/86]*Gamma[17/86]* Gamma[21/86]* Gamma[25/86]*Gamma[41/86])/(Gamma[2/43]*Gamma[8/43]* Gamma[12/43]* Gamma[18/43]*Gamma[20/43]))^(1/10)* ((Gamma[1/43]*Gamma[5/43]*Gamma[9/43]*Gamma[13/43]* Gamma[17/43]* Gamma[21/43])/(Gamma[7/86]*Gamma[15/86]*Gamma[23/86]* Gamma[31/86]*Gamma[39/86]))^(3/10))/(2*86^(7/40)* Pi^(17/20)* Sqrt[Gamma[11/86]]*(Gamma[3/86]*Gamma[19/86])^(7/10)* Gamma[27/86]^(3/5)*Gamma[35/86]^(1/5)* (Gamma[5/86]*Gamma[29/86]*Gamma[33/86]*Gamma[37/86])^(2/5))
Working toward sqrt(163)pi.
It's threatening to be hideous. --rwg
These are purely empirical, unproven results. Incredibly, I'm doing the numerics in Macsyma instead of Mma due to bizarre precision bugs. And bizarreness in general. Floor[<numeric infinite series>] gave no integer. N[%] does, but then N[%] again makes a short float!
But Mma's algebraic number stuff is pretty impressive. Still doesn't denest, tho.
I shouldn't jinx myself, but I think I can do exp(pi sqrt(n/d)) for n and d "within reason". If 163 is beyond reason, wait 'til next year. --rwg
For his ntuplication merit badge, a young Scout asked me how to get the usual factorial ntuplication formula n^(n*z+1/2)*prod((z-i/n)!,i,0,n-1)/(2*%pi)^((n-1)/2) = (n*z)! n - 1 /===\ n z + 1/2 | | i n | | (z - -)! | | n i = 0 ------------------------- = (n z)! n - 1 ----- 2 (2 pi) as the q->1 limit of the q-factorial ntuplication formula (1 - q^n)^(n * z - ((n - 1)/2)) * qpoch(q, q, inf) * product(faq(z - (k/n), q^n),k,0,n - 1)/((1 - q)^(n * z) * qpoch(q^n, q^n, inf)^n) = faq(n * z, q) n - 1 n - 1 n z - ----- /===\ n 2 | | k n (q; q) (1 - q ) | | faq(z - -, q ) inf | | n k = 0 -------------------------------------------------- = faq(n z, q), n n n n z (q ; q ) (1 - q) inf with faq(z,q):=(1-q)^-z*prod((1-q^n)/(1-q^(n+z)),n,1,inf) inf /===\ n | | 1 - q | | ---------- | | z + n n = 1 1 - q faq(z, q) := ---------------- z (1 - q) (The kid's a little slow, and needs to take his shoes off to do transfinite induction.) Using the obvious definition of pi_q, %pi[q] := faq(-1/2,q^2) = sqrt(1-q^2)*qpoch(q^2,q^2,inf)/qpoch(q,q^2,inf) 2 2 2 (q ; q ) sqrt(1 - q ) 1 2 inf pi := faq(- -, q ) = ------------------------ q 2 2 (q; q ) inf I get, for n a power of 2, prod(((q^(2^k) + 1)^(n * z - 2^(k - 1) + 1/2)/(%pi[q^(2^k)])^(2^(k - 1))),k,0,lg(n) - 1)* product(faq(z - (k/n), q^n),k,0,n - 1) = faq(n * z, q) lg(n)-1 k k - 1 n - 1 /===\ 2 n z - 2 + 1/2 /===\ | | (q + 1) | | k n ( | | ---------------------------) | | faq(z - -, q ) = faq(n z, q) | | k - 1 | | n k = 0 2 k = 0 pi k 2 q For general n, |q|<1, the product on the left converges when extended to inifinity, so you can pull the usual hack of dividing two infinite products to get the finite one, but this doesn't say how to actually do the n=3 case: 'limit((1-q^3)^(3*z-1)*qpoch(q,q,inf)/((1-q)^(3*z)*qpoch(q^3,q^3,inf)^3),q,1,minus) = 3^(3*z+1/2)/(2*%pi) 3 3 z - 1 3 z + 1/2 (1 - q ) qpoch(q, q, inf) 3 limit -------------------------------- = ----------. q -> 1- 3 z 3 3 3 2 pi (1 - q) qpoch (q , q , inf) Asking Mma 7.0, In:=Assuming[Abs[q] < 1, Limit[(((1 - q^3)^(3*z - 1)* QPochhammer[q, q])/((1 - q)^(3*z)*(QPochhammer[q^3, q^3])^3)), q -> 1] -> ((3^(3*z + 1/2))/(2*Pi))] "\!\(\* StyleBox[\"\\\"Warning: Contradictory assumption(s) \\\"\", \"MT\"]\)\ \!\(\* StyleBox[ RowBox[{ RowBox[{ RowBox[{\"Abs\", \"[\", \ RowBox[{\"1\", \"+\", \"q\"}], \"]\"}], \"<\", \"1\"}], \"&&\", \ RowBox[{\"0\", \"<\", \"q\", \"<\", FractionBox[\"1\", \ \"1073741824\"]}]}], \"MT\"]\)\!\(\* StyleBox[\"\"\", \"MT\"]\) \ encountered In:=Assuming[Abs[q] < 1, Series[(((1 - q^3)^(3*z - 1)* QPochhammer[q, q])/((1 - q)^(3*z)*(QPochhammer[q^3, q^3])^3)), {q, 1, 2}] -> ((3^(3*z + 1/2))/(2*Pi))] [A very large output was generated. Here is a sample of it:] [. . .] The solution is to write the q-pochhammers as etas, (1-q^n)^(n*z-(n-1)/2)*(q^n)^(n/24)*eta(q)*(prod(faq(z-k/n,q^n),k,0,n-1))/((1-q)^(n*z)*q^(1/24)*eta(q^n)^n) = faq(n*z,q) n - 1 n - 1 n z - ----- /===\ n 2 n n/24 | | k n (1 - q ) (q ) eta(q) | | faq(z - -, q ) | | n k = 0 -------------------------------------------------------- = faq(n z, q) n z 1/24 n n (1 - q) q eta (q ) and use the Jacobi imaginary transformation block([radexpand:false],eta(q),%%=theta_imtrans(%%)) eta(q) = sqrt(-2*%pi/log(q))*eta(%e^(4*%pi^2/log(q))) 2 4 pi ------ 2 pi log(q) eta(q) = sqrt(- ------) eta(%e ) log(q) to get 'limit((1-q^n)^(n*z-(n-1)/2)*(q^n)^(n/24)*eta(q)/((1-q)^(n*z)*q^(1/24)*eta(q^n)^n),q,1,minus) = n^(n*z+1/2)/(2*%pi)^((n-1)/2) n - 1 n z - ----- n 2 n n/24 n z + 1/2 (1 - q ) (q ) eta(q) n limit ----------------------------------- = ----------- q -> 1- n z 1/24 n n n - 1 (1 - q) q eta (q ) ----- 2 (2 pi) This approach is less evident in Mma, whose DedekindEta takes argument tau = log(q)/(2i pi) instead of q. --rwg SELF-EVIDENT FIELD EVENTS
The obvious spacefill of this region takes on a pleasant oriental flavor if you terminate the recursion with quarter-circle arcs: http://gosper.org/erez6.PNG --rwg
DanA> Thanks, Alan.
So, does this make it the same as starting with a plus sign of 5 adjacent squares, and taking 5 congruent copies snuggled together and normalized -- then iterate to the limit? (Actually the boundary of this limit.)
rwg> If you do it right, you can start with a square for Q[0] and get the
"+" sign for Q[1]. See http://gosper.org/erezd.PNG .
These aren't quite the same but they have the same closure and boundary dimension. A toy analogy is the approximations to the unit interval you get by varying Q[0] in the iteration Q[n+1] = (Q[n] U (1+Q[n]))/2. Allan's method, Q[0] := {0}, produces the dyadic rationals. But Q[0] := (0,1] gives the complement of the dyadic rationals. Same closure, utterly unequal measures.
You can't zoom down to a self contact because of self similarity. For greater rigor, you can construct sausage links that will remain demonstrably disjoint. You can also write a function analogous to that obscure Peano Mathematica function that will exactly and continuously map the rationals in the unit interval onto one quarter of the boundary in question. There are also open and closed loop spacefills of this region, which can also be computed exactly for rationals.
I like D=log_5(9) . Any takers on 3D revisitation? --rwg
If so, it's a fractal I was studying just a couple of months ago. I'll go review what I had about it.
=-Dan
<< AllanW> I'm pretty sure I understand the intended construction. We'll construct a sequence of sets of complex numbers; the limit of this sequence will be (well-defined and) the intended fractal.
Let Q[0] contain only 0. Then for any nonnegative integer i, let Q[i+1] = (Q[i] + {0, 1, -1, i, -i}) / (2i + 1). Here, if A and B are sets, the set A+B is intended to mean {a+b | a in A and b in B}.
Each Q is roughly cross-shaped; RWG observes (very tersely) that dividing by 2i+1 rotates and shrinks each such cross by just enough that five crosses can snuggle together to make a meta-cross.
I think this fractal is in Mandelbrot; I cannot dig up my copy at the moment.
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
On Fri, 10 Jul 2009, rwg@sdf.lonestar.org wrote:
The obvious spacefill of this region takes on a pleasant oriental flavor if you terminate the recursion with quarter-circle arcs: http://gosper.org/erez6.PNG --rwg
Beautiful! That'd make a beautiful engraving or inlay pattern for an entryway or furniture piece.
On Fri, 10 Jul 2009, rwg@sdf.lonestar.org wrote:
The obvious spacefill of this region takes on a pleasant oriental flavor if you terminate the recursion with quarter-circle arcs: http://gosper.org/erez6.PNG --rwg
Jason> Beautiful! That'd make a beautiful engraving or inlay pattern for an entryway
or furniture piece.
Thanks! The "floppy" variant, mirror imaging at each level of recursion, is a tad homelier: http://gosper.org/erez7.PNG . But scaled alike, with one maybe mirrored, they make a nice pair. --rwg
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