Plato was right. Pythagoras was right. Pikabu is also right.
Pikabu is a cat. Ze cat. Our cat
Pikabu Ze Cat
ze Cat in HD
Pikabu, while in a "state of happiness", created this table of numbers that she knows about:
Meow!
But now, we, human beings, have the ability to give meaning to symbols. That is the main function of our minds. The more knowledge we have, the more "context" we are able to give to the raw "data".
What meaning can we associate with the numbers above? We know how to multiply numbers and how to add them.
What "magic" is in the above table created by Pikabu ze cat that knows only 0,1,2?
To be continued
P.S.1. Perhaps I should mention that long before Pikabu there were those Babylonians and Hindus
"Reality is that which when you stop believing in it, it doesn't go away."
Philip K. Dick, Valis
I would like to add that Pikabu also can tell time. Rather precisely, too. And if we are late for any of the things that belong to her schedule, she loudly lets us know. And if we don't pay attention immediately, she goes to work on the furniture. That is generally how she gets our attention, except for one thing: when it is time for us to get up in the morning, if we don't heed a few gentle calls, she will jump on the bed and start walking back and forth across our sleeping bodies. Here I should mention that she weighs 8.5 kilos.
ReplyDeleteInteresting is also this: contrary to what I read, that cats are selfish and self-centered, Pikabu is an STO (Service To Others) being. When she talks, she never uses "I" or "me". For instance, when she is hungry, she never cries:
Delete"I am hungry, feed me!!!".
Never! She simply points at her empty dish saying to me:
"Look, this dish is empty. It makes the Universe unbalanced. Are you going to do something about it to restore the balance?"
Interesting.
ReplyDeleteI see three isosceles triangles. Assuming an "element" designates a numerical character in the diagram, Triangle A contains fewer elements than Triangles B and C, which have the same size. If these triangles had infinitely many rows, their sizes would be the same, i.e. countably infinite.
Number of elements read by rows:
Triangle A: 2,3,4,5,6 (natural number progression)
Triangle B: 3,5,7,9,11 (odd number progression)
Triangle C: 3,5,7,9,11 (odd number progression)
Adding the numbers of each row:
Triangle A: 3,3,3,3,3
Triangle B: 4,4,4,4,4
Triangle C: 5,5,5,5,5
Hmm... (3,4,5) the famous Pythagorean triplet!
Multiplying the numbers of each row:
Triangle A: 2,0,0,0,0
Triangle B: 0,0,0,0,0
Triangle C: 4,0,0,0,0
Zero "absorbs" many numbers! The result would have been "richer" if a number other than zero had been used.
Substitution rules (how to obtain Triangle B from A, and C from B):
First row: replace 1 by 20 to get B, then replace rightmost 0 by 1 to get C.
Second row: replace 1 by 200 to get B, then replace rightmost 0 by 1 to get C.
Third row: replace 1 by 2000 to get B, then replace rightmost 0 by 1 to get C.
and so on...
Horizontally, this pattern can be used to create Triangles AA (first row=2220), BB, CC, AAA, BBB, CCC, etc.
Triangle A is fully contained in the right half of Triangle C.
The left half of Triangle B is the left half of Triangle C.
Hence, Triangle C is a mixture of Triangles A and B.
Visually, Triangle B is less appealing because the zeros to the right are not bounded by 1 or 2, as if it was momentarily "imbalanced." So I sense an alternating sequence of "balance-imbalance-balance."
Very interesting observations. I did not notice these regularities.
DeleteLooks like using a cellular automata rule with three different starting states.
ReplyDeleteIndeed, it looks like states of a 3-neighborhood 3-state cellular automaton, like those studied in this paper (see Table 6):
Deletehttps://www.researchgate.net/publication/316251628_Pseudo-random_Number_Generation_using_a_3-state_Cellular_Automaton
The sequence (2,1,2,0,1,2,0,0,1,2,0,0,0,1,2,0,0,0,0,1,...) is obtained when Triangle A is read by rows (from left to right, top to bottom).
ReplyDeleteIf we search for this sequence in The On-line Encyclopedia of Integer Sequences, we find a sequence (https://oeis.org/A135387) which corresponds exactly to the sequence of Triangle A but is represented by a different Triangle:
2;
1, 2;
0, 1, 2;
0, 0, 1, 2;
0, 0, 0, 1, 2;
0, 0, 0, 0, 1, 2;
Interestingly, the eigensequence of this Triangle is equal to the Pell numbers (0, 1, 2, 5, 12, 29,...) (https://oeis.org/A000129)
generated by the following recurrence relation: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
From Wikipedia (https://en.wikipedia.org/wiki/Pell_number):
"In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, 41/29, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82."
"Both the Pell numbers and the companion Pell numbers may be calculated by means of a recurrence relation similar to that for the Fibonacci numbers, and both sequences of numbers grow exponentially, proportionally to powers of the silver ratio 1 + √2. As well as being used to approximate the square root of two, Pell numbers can be used to find square triangular numbers, to construct integer approximations to the right isosceles triangle, and to solve certain combinatorial enumeration problems."
There is also a link with Pythagorean triples:
"As Martin (1875) describes, the Pell numbers can be used to form Pythagorean triples in which a and b are one unit apart, corresponding to right triangles that are nearly isosceles. Each such triple has the form: (2P(n)P(n+1), P^2(n+1) - P^2(n), P^2(n+1) + P^2(n) = P(2n+1))."
"The sequence of Pythagorean triples formed in this way is (4,3,5), (20,21,29), (120,119,169), (696,697,985), …"
Thanks for your comment. I hope to return to Pell numbers later on the series. For a while I am learning this quite ancient number theory. Lot to learn.
Delete