Re: A problem in Pi digits as Lattice space filling
- To: mathgroup at smc.vnet.net
- Subject: [mg94160] Re: A problem in Pi digits as Lattice space filling
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Sat, 6 Dec 2008 06:14:12 -0500 (EST)
- References: <ggj7co$j2i$1@smc.vnet.net> <200811261222.HAA22459@smc.vnet.net>
- Reply-to: drmajorbob at longhorns.com
A normal number has uniformly distributed digits (not normally distributed ones). But the converse fails. Take, for example, the rational number 0.12345678901234567890123456789 // Rationalize 137174210/1111111111 whose digits are uniformly distributed (whatever that actually means for NONRANDOM numbers!), but is clearly not a "normal number" since, for instance, "13" does not appear as a subsequence with frequency 1/100. Bobby On Wed, 03 Dec 2008 05:32:34 -0600, dh <dh at metrohm.com> wrote: > Hi Bob, > it is not known, but at least the first 30 million digits are pretty > uniform. Further, there is a hugh pitfall here, a number with normal > distributed digits is called normal!! See e.g.: > http://mathworld.wolfram.com/PiDigits.html > Daniel > > DrMajorBob wrote: >> It does LOOK uniform, I admit. I just wonder whether it's a known fact. >> Bobby >> On Thu, 27 Nov 2008 07:17:54 -0600, dh <dh at metrohm.com> wrote: >> >>> Hi Bob, >>> try: >>> d = RealDigits[Pi, 10, 10^6][[1]]; >>> and make a histogram of it. Shure this is no proof, but it does >>> certainly not look normal. >>> Daniel >>> >>> >>> DrMajorBob wrote: >>>>> first some picky things. The digits in Pi are not normal, but >>>>> uniformly >>>>> distributed >>>> I doubt that (uniformity) is actually known. Is it? >>>> Bobby >>>> On Wed, 26 Nov 2008 06:22:10 -0600, dh <dh at metrohm.com> wrote: >>>>> >>>>> Hi Roger, >>>>> >>>>> first some picky things. The digits in Pi are not normal, but >>>>> uniformly >>>>> >>>>> distributed. Further you should have mentioned that you define the >>>>> >>>>> coordinate tuples by moving only one digit. Then, I can verify the >>>>> >>>>> filling number for 1 dim. is 33, but for 2 dim. with 606 the duple >>>>> {6,8} >>>>> >>>>> does not appear, you need 607 digits. For 3 dim. I need 8556, not >>>>> 8554. >>>>> >>>>> Finally, here is a way to do it rather fast. I give the example for >>>>> 3 dim: >>>>> >>>>> n=8556; >>>>> >>>>> all=Flatten[Table[{i1,i2,i3},{i1,0,9},{i2,0,9},{i3,0,9}],2]; >>>>> >>>>> d=Union@Partition[RealDigits[\[Pi],10,n][[1]],3,1]; >>>>> >>>>> Complement[all,d] >>>>> >>>>> You increase n until the answer is empty: {}. Of course this can be >>>>> >>>>> automated by wrapping a binary search around the code above, but I >>>>> let >>>>> >>>>> this for you. >>>>> >>>>> hope this helps, Daniel >>>>> >>>>> >>>>> >>>>> >>>>> >>>>> Roger Bagula wrote: >>>>> >>>>>> I need help with programs for 4th, 5th and 6th, etc. >>>>>> levels of lattice filling: >>>>>> The idea that the Pi digits are normal >>>>>> implies that they will fill space on different levels >>>>>> in a lattice type way ( Hilbert/ Peano space fill). >>>>>> Question of space filling: >>>>>> Digit(n)-> how fast till all ten >>>>>> {Digit[n],Digit[n+1]} -> how fast to fill the square lattice >>>>>> {0,0},{0,9},{9,0},{9,9} >>>>>> {Digit[n],Digit[n+1],Digit[n+2]} -> how fast to fill the cubic >>>>>> lattice >>>>>> {0,0,0} to {9,9,9} >>>>>> I've answers for the first three with some really clunky programs. >>>>>> 33,606,8554,... >>>>>> my estimates for the 4th is: 60372 to 71947 >>>>>> ((8554)2/(606*2)and half the log[]line result of 140000.) >>>>>> but it appears to outside what my old Mac can do. >>>>>> I'd also like to graph the first occurrence to see how random the >>>>>> path >>>>>> between >>>>>> the lattice / space fill points is. >>>>>> The square: >>>>>> a = Table[Floor[Mod[N[Pi*10^n, 1000], 10]], {n, 0, 1000}]; >>>>>> Flatten[Table[ >>>>>> If[Length[Delete[Union[Flatten[Table[Table[If[a[[n]] == k && >>>>>> a[[n + \ >>>>>> 1]] - l == >>>>>> 0, {l, k}, {}], {k, 0, 9}, { >>>>>> l, 0, 9}], {n, 1, m}], 2]], 1]] == 100, m, {}], {m, 600, >>>>>> 610}]] >>>>>> {606, 607, 608, 609, 610} >>>>>> Table[Length[Delete[Union[Flatten[Table[Table[If[a[[n]] == k && a[[ >>>>>> n + 1]] - l == 0, {l, k}, {}], {k, 0, 9}, {l, >>>>>> 0, 9}], {n, 1, m}], 2]], 1]], {m, 600, 650}] >>>>>> {99, 99, 99, >>>>>> 99, 99, 99, 100, >>>>>> 100, 100, 100, >>>>>> 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, >>>>>> 100, 100, >>>>>> 100, 100, >>>>>> 100, 100, 100, >>>>>> 100, 100, 100, 100, 100, >>>>>> 100, 100, 100, >>>>>> 100, 100, 100, 100, 100, >>>>>> 100, 100, 100, 100, 100, 100, 100, 100, 100} >>>>>> Mathematica >>>>>> Clear[a, b, n] >>>>>> a = Table[Floor[Mod[N[Pi*10^n, 10000], 10]], {n, 0, 10000}]; >>>>>> b = Table[{a[[n]], a[[n + 1]], a[[n + 2]]}, {n, 1, Length[a] - 2}]; >>>>>> Flatten[Table[ >>>>>> If[Length[Union[Table[b[[n]], { >>>>>> n, 1, m}]]] == 1000, m, {}], {m, 8550, 8600}]] >>>>>> {8554, 8555, 8556, >>>>>> 8557, 8558, 8559, 8560, 8561, 8562, 8563, 8564, 8565, 8566, >>>>>> 8567, 8568, 8569, 8570, 8571, 8572, 8573, 8574, 8575, 8576, >>>>>> 8577, 8578, 8579, 8580, 8581, 8582, 8583, 8584, 8585, >>>>>> 8586, >>>>>> 8587, 8588, 8589, >>>>>> 8590, 8591, 8592, 8593, 8594, 8595, 8596, 8597, 8598, 8599, >>>>>> 8600} >>>>>> The proof of the 33 is: >>>>>> Clear[a,b,n] >>>>>> a=Table[Floor[Mod[N[Pi*10^n,10000],10]],{n,0,10000}]; >>>>>> Flatten[Table[If[Length[Union[Table[a[[n]],{n,1, >>>>>> m}]]]==10,m,{}],{m,1,50}]] >>>>>> {33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, >>>>>> 49, 50} >>>>>> My own 4th level program that won't run on my older machine /older >>>>>> version of Mathematica: >>>>>> Clear[a, b, n] >>>>>> a = Table[Floor[Mod[N[Pi*10^n, 100000], 10]], {n, 0, 100000}]; >>>>>> b = Table[{a[[n]], a[[n + 1]], a[[n + 2]], a[[n + 3]]}, {n, 1, >>>>>> Length[a] >>>>>> - 3}]; >>>>>> Flatten[Table[If[ Length[Union[Table[b[[n]], {n, 1, m}]]] == 10000, >>>>>> m, >>>>>> {}], {m, 1, 50}]] >>>>>> Respectfully, Roger L. Bagula >>>>>> 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 >>>>>> :http://www.geocities.com/rlbagulatftn/Index.html >>>>>> alternative email: rlbagula at sbcglobal.net >>>>> >>>>> >>>> >>> >>> >> > > -- DrMajorBob at longhorns.com