Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca - contact me Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca on Twitter Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca - Lumondo Photography Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca - Pi Art Martin Krzywinski / Genome Sciences Center / mkweb.bcgsc.ca - Hilbertonians - Creatures on the Hilbert Curve
Embrace me, surround me as the rush comes.Motorcycledrift deeper into the soundmore quotes

numbers: exciting


EMBO Practical Course: Bioinformatics and Genome Analysis, 5–17 June 2017.


visualization + design

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The 2017 Pi Day art imagines the digits of Pi as a star catalogue with constellations of extinct animals and plants. The work is featured in the article Pi in the Sky at the Scientific American SA Visual blog.

`\pi` Day 2014 Art Posters


Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2017 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2016 `\pi` approximation day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2016 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2015 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2014 `\pi` approx day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2014 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
2013 `\pi` day

Pi Day 2014 Art Poster - Folding the Number Pi
 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Circular `\pi` art

On March 14th celebrate `\pi` Day. Hug `\pi`—find a way to do it.

For those who favour `\tau=2\pi` will have to postpone celebrations until July 26th. That's what you get for thinking that `\pi` is wrong.

If you're not into details, you may opt to party on July 22nd, which is `\pi` approximation day (`\pi` ≈ 22/7). It's 20% more accurate that the official `\pi` day!

Finally, if you believe that `\pi = 3`, you should read why `\pi` is not equal to 3.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
All art posters are available for purchase.
I take custom requests.

For the 2014 `\pi` day, two styles of posters are available: folded paths and frequency circles.

The folded paths show `\pi` on a path that maximizes adjacent prime digits and were created using a protein-folding algorithm.

The frequency circles colourfully depict the ratio of digits in groupings of 3 or 6. Oh, look, there's the Feynman Point!

Explore the Feynman Point and other such artefacts in `\pi` using René Hansen's interactive version of this year's posters.

The Feynman Point

The Feynman Point is the first place where 6 9s occur in π . This happens at digit 762, which is much sooner than expected. Here is π to 1,000 decimal places showing the location of the Feynman point.

3.1415926535 8979323846 2643383279 5028841971 6939937510  : 50
  5820974944 5923078164 0628620899 8628034825 3421170679  : 100
  8214808651 3282306647 0938446095 5058223172 5359408128  : 150
  4811174502 8410270193 8521105559 6446229489 5493038196  : 200
  4428810975 6659334461 2847564823 3786783165 2712019091  : 250
  4564856692 3460348610 4543266482 1339360726 0249141273  : 300
  7245870066 0631558817 4881520920 9628292540 9171536436  : 350
  7892590360 0113305305 4882046652 1384146951 9415116094  : 400
  3305727036 5759591953 0921861173 8193261179 3105118548  : 450
  0744623799 6274956735 1885752724 8912279381 8301194912  : 500

  9833673362 4406566430 8602139494 6395224737 1907021798  : 550
  6094370277 0539217176 2931767523 8467481846 7669405132  : 600
  0005681271 4526356082 7785771342 7577896091 7363717872  : 650
  1468440901 2249534301 4654958537 1050792279 6892589235  : 700
  4201995611 2129021960 8640344181 5981362977 4771309960  : 750
  5187072113 4999999837 2978049951 0597317328 1609631859  : 800
  5024459455 3469083026 4252230825 3344685035 2619311881  : 850
  7101000313 7838752886 5875332083 8142061717 7669147303  : 900
  5982534904 2875546873 1159562863 8823537875 9375195778  : 950
  1857780532 1712268066 1300192787 6611195909 2164201989  : 1000

As the story goes, Feynman said that he'd like to memorize π up to the first 6 9s so that he could finish the recitation with "nine nine nine nine nine nine and so on". To find a scheme that suggests that π is rational is mathematical humor of the highest order.

We can ask a general question — where is the first place where digit d occurs n times? For example, where do we see the first 5555 (d=5,n=4) or 777777 (d=7,n=6) .

I look into these kinds of locations in π below. If you are interested in integer sequences in general, see the on-line encyclopedia of integer sequences (OEIS)

(d,n) points in π — sequences of repeating digits

I call a location at which digit d is seen n times in a row the (d,n) point. The Feynman point is a specific case — it is a (d=9,n=6) point.

One can talk about all the (d,n) points (i.e. (d,n,0) , (d,n,1) , (d,n,2) ...), which is the set of all locations at which d appears n times. It should be clear from the context whether one point is being referenced, or a set of points.

We can add a location index i to refer to a specific instance of the sequence. The ith appearance of the digit sequence is (d,n,i) . If we're talking about a single point and i is not specified then i=0 is assumed, which refers to the first appearance of the sequence for a given d and n. That is, (d,n) = (d,n,i=0) .

The point (d,1) trivially corresponds to the first occurence of digit d.

Below is a list of all (d,n=6,i=0) points.

 d n  i sequence position .....(n x d) .....
 1 6  0   111111   255945 62417(111111)75895
 2 6  0   222222   963024 87437(222222)85444
 4 6  0   444444   828499 02846(444444)66922
 5 6  0   555555   244453 33233(555555)69581
 6 6  0   666666   252499 58934(666666)88391
 7 6  0   777777   399579 18074(777777)98344
 8 6  0   888888   222299 10985(888888)35254
 9 6  0   999999      762 21134(999999)83729 # Feynman Point

examples of (d,n) points in first 1,000,000 digits of π

There are 49 (d,n,i=0) points in the first 1,000,000 digits of π for n>1. For example, the first (d=0,n=4) , which is the first 0000, occurs at digit 13,390 (...93095000090715). The longest sequence of d=0 happens for n=7(d=0,n=7,i=0) is at position 710,100 (...73537333333386381).

 d n  i    n x d position ...(n x d)...
 0 2  0       00      307 24587(00)66063
 1 2  0       11       94 25342(11)70679
 2 2  0       22      135 55058(22)31725
 3 2  0       33       24 46264(33)83279
 4 2  0       44       59 09749(44)59230
 5 2  0       55      130 44609(55)05822
 6 2  0       66      117 28230(66)47093
 7 2  0       77      559 43702(77)05392
 8 2  0       88       34 79502(88)41971
 9 2  0       99       44 71693(99)37510
 0 3  0      000      601 05132(000)56812
 1 3  0      111      153 12848(111)74502
 2 3  0      222     1735 99219(222)18427
 3 3  0      333     1698 39414(333)45477
 4 3  0      444     2707 42858(444)79526
 5 3  0      555      177 52110(555)96446
 6 3  0      666     2440 49954(666)72782
 7 3  0      777     4575 94764(777)26224
 8 3  0      888     4985 08099(888)68741
 9 3  0      999     2949 19217(999)83910
 0 4  0     0000    13390 93095(0000)90715
 1 4  0     1111    12700 42144(1111)26358
 2 4  0     2222     4902 35136(2222)47715
 3 4  0     3333    66846 31487(3333)67147
 4 4  0     4444    54525 11793(4444)82014
 5 4  0     5555    33172 96839(5555)68686
 6 4  0     6666    21880 23047(6666)72174
 7 4  0     7777     1589 28909(7777)27938
 8 4  0     8888     4751 16274(8888)00786
 9 4  0     9999    17988 98955(9999)11209
 0 5  0    00000    17534 66768(00000)10652
 1 5  0    11111    32788 46584(11111)57758
 2 5  0    22222    65260 99725(22222)80801
 3 5  0    33333    28467 52392(33333)64743
 4 5  0    44444   808650 64849(44444)73161
 5 5  0    55555    24466 89742(55555)16076
 6 5  0    66666    48439 71972(66666)42267
 7 5  0    77777   162248 47779(77777)18415
 8 5  0    88888   213245 09581(88888)03131
 9 5  0    99999    19446 21285(99999)39961
 1 6  0   111111   255945 62417(111111)75895
 2 6  0   222222   963024 87437(222222)85444
 4 6  0   444444   828499 02846(444444)66922
 5 6  0   555555   244453 33233(555555)69581
 6 6  0   666666   252499 58934(666666)88391
 7 6  0   777777   399579 18074(777777)98344
 8 6  0   888888   222299 10985(888888)35254
 9 6  0   999999      762 21134(999999)83729
 3 7  0  3333333   710100 73537(3333333)86381

When we sort this list by position, we can see just how unusual (i.e. early) the Feynman Point is. We also see (d,n=2) , (d,n=3) , (d,n=4) and (d,n=5) points for all digits d) in the first 1,000,000 digits.

 3 2  0       33       24 46264(33)83279      # first n=2
 8 2  0       88       34 79502(88)41971
 9 2  0       99       44 71693(99)37510
 4 2  0       44       59 09749(44)59230
 1 2  0       11       94 25342(11)70679
 6 2  0       66      117 28230(66)47093
 5 2  0       55      130 44609(55)05822
 2 2  0       22      135 55058(22)31725
 1 3  0      111      153 12848(111)74502     # first n=3
 5 3  0      555      177 52110(555)96446
 0 2  0       00      307 24587(00)66063
 7 2  0       77      559 43702(77)05392      #  last n=2
 0 3  0      000      601 05132(000)56812
 9 6  0   999999      762 21134(999999)83729  # first n=6 (Feynman Point)
 7 4  0     7777     1589 28909(7777)27938    # first n=4
 3 3  0      333     1698 39414(333)45477
 2 3  0      222     1735 99219(222)18427
 6 3  0      666     2440 49954(666)72782
 4 3  0      444     2707 42858(444)79526
 9 3  0      999     2949 19217(999)83910
 7 3  0      777     4575 94764(777)26224
 8 4  0     8888     4751 16274(8888)00786
 2 4  0     2222     4902 35136(2222)47715
 8 3  0      888     4985 08099(888)68741     #  last n=3
 1 4  0     1111    12700 42144(1111)26358
 0 4  0     0000    13390 93095(0000)90715
 0 5  0    00000    17534 66768(00000)10652   # first n=5
 9 4  0     9999    17988 98955(9999)11209
 9 5  0    99999    19446 21285(99999)39961
 6 4  0     6666    21880 23047(6666)72174
 5 5  0    55555    24466 89742(55555)16076
 3 5  0    33333    28467 52392(33333)64743
 1 5  0    11111    32788 46584(11111)57758
 5 4  0     5555    33172 96839(5555)68686
 6 5  0    66666    48439 71972(66666)42267
 4 4  0     4444    54525 11793(4444)82014
 2 5  0    22222    65260 99725(22222)80801
 3 4  0     3333    66846 31487(3333)67147    #  last n=4
 7 5  0    77777   162248 47779(77777)18415  
 8 5  0    88888   213245 09581(88888)03131
 8 6  0   888888   222299 10985(888888)35254
 5 6  0   555555   244453 33233(555555)69581
 6 6  0   666666   252499 58934(666666)88391
 1 6  0   111111   255945 62417(111111)75895
 7 6  0   777777   399579 18074(777777)98344
 3 7  0  3333333   710100 73537(3333333)86381 # first n=7
 4 5  0    44444   808650 64849(44444)73161   #  last n=5
 4 6  0   444444   828499 02846(444444)66922
 2 6  0   222222   963024 87437(222222)85444

when do we expect (d,n) points?

Any given position in π , 10n different combinations of n digits can appear. In only one of them are all the digits the same and equal to d. Thus, the chance of seeing (d,n) at any position for any given d is p=10n.

The probability of not seeing a (d,n) point at a location for a given d is 1-p.

Thus the probability of not seeing a point at k locations and then seeing one at k+1 location is (1-p)kp. This is the definition of the geometric distribution.

For example, the question "how many times must we throw a die until a 1 appears?" can be addressed by this distribution. Here p=1/6, assuming the die is fair. The probability of seeing a 1 for the first time on the k toss is

k   P(X=k)
1   16.7% = (1-1/6)0 * 1/6
2   13.8% = (1-1/6)1 * 1/6 
3   11.6% = (1-1/6)2 * 1/6  
4    9.6% = (1-1/6)3 * 1/6 
5    6.7% = (1-1/6)4 * 1/6 
6    5.6% = (1-1/6)5 * 1/6 
...

The cumulative probability P(Xk) gives us the chance of seeing a 1 for the first time on the 1, 2, 3 ... or k toss). 1-P(Xk) is the chance of not seeing a 1 in the first 1, 2, 3 ... k tosses.

k   P(X≤k)   1-P(X≤k)
1   16.7%    83.3%
2   30.6%    69.4%
3   42.1%    57.9%
4   51.8%    48.2%
5   59.8%    40.2%
6   66.5%    33.5%
...

We can apply this calculation to the probability of observing a (d,n) point for the first time. Let's pick (d=9,n=6) . Using p=10–6, let's look at 1-P(Xk). Values in the table below for P are shown to one significant digit, except for the Feynman Point position k=962.

k      P(X≤k)     1-P(X≤k)    
1      0.000001   0.999999
10     0.00001    0.99999
100    0.0001     0.9999
962    0.000962   0.999037
1000   0.001      0.999
10000  0.01       0.99
100000 0.1        0.9
693147 0.5        0.5

Now we know just how unlikely the Feynman Point is. The chance of observing 6 9s in the first 962 digits is about 0.000962.

These probabilities are approximate because probability that a sequence of n digits appears at any given position is correlated across a window of n-1 digits. For example if we see (d=9,6) at position j then the probability of seeing (d=9,6) at position j+1 is 1/10, not 10–6, because we are guaranteed the first 5 9s.

all (d,n) points in 268 million digits of π

If you're interested in more (d,n) points, I calculated all for lengths n≥3 for the first ~268 million digits of π — you can download the complete list of the 2.42 million (d,n) points.

Most of these points are (d,n=3) (2.17 million). The count as function of n is

n  number of (d,n) points
3  2174839
4  217431
5  21749
6  2153
7  204
8  19
9  3

The Feynman Point is (d=9,n=6,i=0) . With 268 million digits, we can find up to (d=9,n=8,i=0) — the first places in which 8 9's appear in a row. All the (d=9,n,i=9) points in the first 268 million digits of π are

 9 3  0      999     2949 19217(999)83910
 9 4  0     9999    17988 98955(9999)11209
 9 5  0    99999    19446 21285(99999)39961
 9 6  0   999999      762 21134(999999)83729
 9 7  0  9999999  1722776 09713(9999999)31766
 9 8  0 99999999 36356642 15746(99999999)54228

The longest sequences are for n=8 and 9, of which there are 50. We see a (d,n=8,0) point for all d.

 0 8  0 00000000 172330850 65581(00000000)12202
 0 8  1 00000000 184688988 50614(00000000)27944
 1 8  0 11111111 159090113 46225(11111111)09751
 1 8  1 11111111 174624972 36529(11111111)76023
 1 8  2 11111111 199394968 58734(11111111)25842
 2 8  0 22222222 175820910 95134(22222222)07695
 3 8  0 33333333 36488176 81791(33333333)25108
 3 8  1 33333333 248922246 51334(33333333)47253
 4 8  0 44444444 22931745 74369(44444444)36403
 4 8  1 44444444 65122865 35213(44444444)82112
 4 8  2 44444444 221749424 95631(44444444)79916
 5 8  0 55555555 168743355 75041(55555555)01882
 6 8  0 66666666 55616210 64263(66666666)81935
 6 8  1 66666666 129423072 80429(66666666)81580
 6 8  2 66666666 160301327 23994(66666666)89064
 7 8  0 77777777 82144203 84454(77777777)83966
 8 8  0 88888888 239798471 04446(88888888)16169
 9 8  0 99999999 36356642 15746(99999999)54228
 9 8  1 99999999 66780105 26137(99999999)31798
 6 9  0 666666666 45681781 79094(666666666)71734
 7 9  0 777777777 24658601 85304(777777777)24846
 8 9  0 888888888 46663520 87842(888888888)07509

subsequences of π in 268 million digits of π

Finally, it's interesting to see where subsequences of π occurs in π . Because π is random and non-terminating, it has infinitely many subsequences of itself in it — this is hard to think about.

In the first 268 million digits of π you see subsequences up to 8 digits, not counting the trivial one that starts at zero.

       3 1  0        9 59265(3)58979
      31 2  0      137 05822(31)72535
     314 3  0     2120 96514(314)29809
    3141 4  0     3496 67110(3141)26711
   31415 5  0    88008 96265(31415)14138
  314159 6  0   176451 24573(314159)78761
 3141592 7  0 25198140 60173(3141592)14513
31415926 8  0 50366472 17005(31415926)09521

We can also find e (2.718281828) up to 9 digits in the first 268 digits of π .

       2  1  0        6 14159(2)65358
      27  2  0       28 43383(27)95028
     271  3  0      241 83165(271)20190
    2718  4  0    11706 24322(2718)85159
   27182  5  0    28024 10077(27182)71874
  271828  6  0    33789 50445(271828)92749
 2718281  7  0  1526800 07421(2718281)53375
27182818  8  0 73154827 92454(27182818)56113
271828182 9  0 246890641 99305(271828182)91617

as well as φ (1.618033989) up to 8 digits,

       1 1  0        1 3(1)41592
      16 2  0       40 84197(16)93993
     161 3  0     1610 70600(161)45249
    1618 4  0     6004 60290(1618)76679
   16180 5  0   105857 23121(16180)99462
  161803 6  0   144979 54397(161803)03956
 1618033 7  0  1205122 51502(1618033)21817
16180339 8  0 19445230 33490(16180339)99933

as well as 1/ φ = 1– φ up to 6 digits.

       0 1  0        32 32795(0)28841
      06 2  0        71 78164(06)28620
     061 3  0       885 38142(061)71776
    0618 4  0     14423 26086(0618)72455
   06180 5  0     84671 08181(06180)21002
  061803 6  0    933127 07779(061803)00371
 0618033 7  0  12074179 53993(0618033)75138
06180339 8  0 200380161 62492(06180339)55938

VIEW ALL

news + thoughts

Tabular Data

Tue 11-04-2017
Tabulating the number of objects in categories of interest dates back to the earliest records of commerce and population censuses.

After 30 columns, this is our first one without a single figure. Sometimes a table is all you need.

In this column, we discuss nominal categorical data, in which data points are assigned to categories in which there is no implied order. We introduce one-way and two-way tables and the `\chi^2` and Fisher's exact tests.

Altman, N. & Krzywinski, M. (2017) Points of Significance: Tabular data. Nature Methods 14:329–330.

...more about the Points of Significance column

Happy 2017 `\pi` Day—Star Charts, Creatures Once Living and a Poem

Tue 14-03-2017


on a brim of echo,

capsized chamber
drawn into our constellation, and cooling.
—Paolo Marcazzan

Celebrate `\pi` Day (March 14th) with star chart of the digits. The charts draw 40,000 stars generated from the first 12 million digits.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
12,000,000 digits of `\pi` interpreted as a star catalogue. (details)

The 80 constellations are extinct animals and plants. Here you'll find old friends and new stories. Read about how Desmodus is always trying to escape or how Megalodon terrorizes the poor Tecopa! Most constellations have a story.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Find friends and stories among the 80 constellations of extinct animals and plants. Oh look, a Dodo guardings his eggs! (details)

This year I collaborate with Paolo Marcazzan, a Canadian poet, who contributes a poem, Of Black Body, about space and things we might find and lose there.

Check out art from previous years: 2013 `\pi` Day and 2014 `\pi` Day, 2015 `\pi` Day and and 2016 `\pi` Day.

Data in New Dimensions: convergence of art, genomics and bioinformatics

Tue 07-03-2017

Art is science in love.
— E.F. Weisslitz

A behind-the-scenes look at the making of our stereoscopic images which were at display at the AGBT 2017 Conference in February. The art is a creative collaboration with Becton Dickinson and The Linus Group.

Its creation began with the concept of differences and my writeup of the creative and design process focuses on storytelling and how concept of differences is incorporated into the art.

Oh, and this might be a good time to pick up some red-blue 3D glasses.

BD Genomics 3D art exhibit - AGBT 2017 / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
A stereoscopic image and its interpretive panel of single-cell transcriptomes of blood cells: diseased versus healthy control.

Interpreting P values

Thu 02-03-2017
A P value measures a sample’s compatibility with a hypothesis, not the truth of the hypothesis.

This month we continue our discussion about `P` values and focus on the fact that `P` value is a probability statement about the observed sample in the context of a hypothesis, not about the hypothesis being tested.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Interpreting P values. (read)

Given that we are always interested in making inferences about hypotheses, we discuss how `P` values can be used to do this by way of the Benjamin-Berger bound, `\bar{B}` on the Bayes factor, `B`.

Heuristics such as these are valuable in helping to interpret `P` values, though we stress that `P` values vary from sample to sample and hence many sources of evidence need to be examined before drawing scientific conclusions.

Altman, N. & Krzywinski, M. (2017) Points of Significance: Interpreting P values. Nature Methods 14:213–214.

Background reading

Krzywinski, M. & Altman, N. (2017) Points of significance: P values and the search for significance. Nature Methods 14:3–4.

Krzywinski, M. & Altman, N. (2013) Points of significance: Significance, P values and t–tests. Nature Methods 10:1041–1042.

...more about the Points of Significance column

Snellen Charts—Typography to Really Look at

Sat 18-02-2017

Another collection of typographical posters. These ones really ask you to look.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Snellen charts designed using physical constants, Braille and elemental abundances in the universe and human body.

The charts show a variety of interesting symbols and operators found in science and math. The design is in the style of a Snellen chart and typset with the Rockwell font.

Essentials of Data Visualization—8-part video series

Fri 17-02-2017
Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca

In collaboration with the Phil Poronnik and Kim Bell-Anderson at the University of Sydney, I'm delighted to share with you our 8-part video series project about thinking about drawing data and communicating science.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Essentials of Data Visualization: Thinking about drawing data and communicating science.

We've created 8 videos, each focusing on a different essential idea in data visualization: encoding, shapes, color, uncertainty, design, drawing missing or unobserved data, labels and process.

The videos were designed as teaching materials. Each video comes with a slide deck and exercises.