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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


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