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Wolfram Summer School

June 24–July 14, 2018
Bentley University, Waltham, MA


Andrius Kulikauskas

Class of 2008


Andrius Kulikauskas was born in 1964 in Santa Monica, California. In 1997 he moved to Lithuania, the land of his heritage. There he started Minciu Sodas, an online laboratory for independent thinkers around the world. His quest in life is to know everything and apply that knowledge usefully. His deepest value is living by truth.

His PhD thesis, in mathematics and algebraic combinatorics, is Symmetric Functions of the Eigenvalues of a Matrix.

Project: Salingaros's Laws of Architecture

Is beauty real?

This project will bridge the work of Stephen Wolfram and Christopher Alexander. Nikos Salingaros is the editor of Christopher Alexander's volumes The Nature of Order and the author of The Theory of Architecture, in which he formulates three laws of architecture that are basic to Alexander's fifteen principles of life. Each law explains how the optimal architectural environment helps us be sensitive and responsive, and each law has the environment which allows us to differentiate it as much as possible:

  • Locally there should be simple alternation of atoms (such as black and white tiles)
  • Globally the system should be as deliberate as possible, with minimal entropy
  • There should be as many levels of scale as possible, but not closer than e=2.718.

The main technical goal of this project is to find a set of cellular automata that yields a set of optimal patterns. This led to considering rule 105 which is an ordinary 3-cell, 2-color, class 2 automaton. It has several repetitive backgrounds, called wallpapers. These wallpapers obey rules for combination, giving a scale. These combinations quickly generate pleasing patterns that most people would call beautiful.

Favorite Radius 3/2 Rule

Rule 34695

For each automata I calculated two parameters that helped me find interesting cases.

The first parameter is inspired by Christopher Langton's work, which itself was inspired by Stephen Wolfram. Christopher calculated the ratio of "live" and "dead" cells and we can likewise for "black" and "white" cells. He observed that the "random," "boiling" activity of class 3 was expected when the ratio was 1:1, whereas class 4 behavior would occur a bit away from that, closer to the simpler class 2 and class 1 behavior, I suppose because class 4 must emulate both kinds. Michael Schreiber encouraged me to look at the de Bruijn sequence as the initial condition, which I wrapped around itself and then, as he suggested, counted how many unique outputs it would create going down (where two patterns are considered the same if one can be rotated into the other). Then I calculated the ratios of the black and white cells among the outputs. Many automata had a 1:1 relation but, being curious, I looked for the automata with a ratio that was closest to 1:1 but not exactly equal to that. This rule 34695 looks just like rule 30. It has black-to-white ratio of 81:79. Kerry Alley noticed that it included a reflection but that otherwise the shape was the same. She and Jason Cawley analyzed the rule and explained how it is almost the same as rule 30 with one inactive bit (which is rule 7710) except for the reflection, which is possible in this 3/2 space but not in the simpler space. So these are curious coincidences and so I choose rule 34695 as my favorite rule and I also note its flip, rule 22102.

The second parameter is simply the number of steps that it takes for the de Bruijn sequence (wrapped around itself so that it goes down a cylinder) to go down before repeating (perhaps under rotation). I ordered the automata by this number. The highest number was 23. This is an effective way to discover many interesting automata including those which seem to exhibit class 4 behavior, such as: 37621, 38554, 39502, 42968, 9510, 9817, 19766, 26329, 38264, 46227, 50537, 51975, 15564, 17531, 18337, 23718, 27108, 39340, 47705, 54330, 57367, 57686, 1446, 7499, 10633, 12050, 14646, 18778, 23846, 24241, 34776, which all appeared within the first 300 in the sequence. Also, it caught automata whose output is trivial under a single dot.