DM42 OFF Images

[Joe Horn]

(06-10-2022 07:16 AM)LinusSch Wrote:  

(06-10-2022 06:50 AM)Joe Horn Wrote:  I had HOPED that they would look at least slightly different from each other, but to my eye they all look like utterly random noise.

This is a great concept, however, all irrational numbers should look like utterly random noise when inspected like this. With rational numbers there should be patterns.

I hope I'm not being pedantic by disagreeing. Rational numbers *can* have patterns in their decimal (or any other base) expansion. They just can't have *repeating* patterns. The example usually given is 1.010010001000010000010000001... where each string of zeros contains one more zero than the previous string. That's a definite pattern, but it never repeats, so the overall number is irrational. Furthermore, the word "repeating" must be clarified, since repeated strings of 3 zeros between all the digits of pi (3.00010004000100050009...) has a "repeated pattern" of 3 zeros, but they're interrupted by the digits of pi, so the overall number again is irrational.

Therefore, images whose pixels represent the bits of a binary irrational number *can* contain eye-catching patterns. The 11 values I chose above just didn't happen to do that.

(06-10-2022 07:16 AM)LinusSch Wrote:  So let me suggest a slight tweak: create these images for increasingly accurate rational approximations of pi, ending with the exact one already created, for a set of images with hopefully interesting differences!

Intriguing! It will be far easier to get patterns this way, since the partial quotients of many irrational numbers *do* follow patterns, e.g. the constant e whose continued fraction expansion is {2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12...} and the continued fraction for tan(1 radian) is {1;1,1,3,1,5,1,7,1,9,1,11...} which are not *repeating* patterns, but they are definitely recognizable and extendable patterns. So the task at hand is to find ones whose pattern looks good in binary. Thanks for the delightful challenge!