Crikey Math

Having fun exploring mathematics

  • Home
  • About
  • Blog
  • Contribute
en English
af Afrikaanssq Albanianam Amharicar Arabichy Armenianaz Azerbaijanieu Basquebe Belarusianbn Bengalibs Bosnianbg Bulgarianca Catalanceb Cebuanony Chichewazh-CN Chinese (Simplified)zh-TW Chinese (Traditional)co Corsicanhr Croatiancs Czechda Danishnl Dutchen Englisheo Esperantoet Estoniantl Filipinofi Finnishfr Frenchfy Frisiangl Galicianka Georgiande Germanel Greekgu Gujaratiht Haitian Creoleha Hausahaw Hawaiianiw Hebrewhi Hindihmn Hmonghu Hungarianis Icelandicig Igboid Indonesianga Irishit Italianja Japanesejw Javanesekn Kannadakk Kazakhkm Khmerko Koreanku Kurdish (Kurmanji)ky Kyrgyzlo Laola Latinlv Latvianlt Lithuanianlb Luxembourgishmk Macedonianmg Malagasyms Malayml Malayalammt Maltesemi Maorimr Marathimn Mongolianmy Myanmar (Burmese)ne Nepalino Norwegianps Pashtofa Persianpl Polishpt Portuguesepa Punjabiro Romanianru Russiansm Samoangd Scottish Gaelicsr Serbianst Sesothosn Shonasd Sindhisi Sinhalask Slovaksl Slovenianso Somalies Spanishsu Sudanesesw Swahilisv Swedishtg Tajikta Tamilte Teluguth Thaitr Turkishuk Ukrainianur Urduuz Uzbekvi Vietnamesecy Welshxh Xhosayi Yiddishyo Yorubazu Zulu

Exploding dots: 1 <- 2 and 2<- 4

October 10, 2017 By Gary Ernest Davis

Exploding dots machines arise from patterns of dot replacement, which should, in turn, determine a “base”.

For example the replacement pattern 1 \leftarrow 2 gives rise to a base b satisfying 1\times b = 2 , whereas the replacement pattern 2 \leftarrow 3 gives rise to a rational base b satisfying 2\times b = 3 .

The replacement  pattern 1\vert0\vert0 \leftarrow 0\vert 1\vert 1 gives rise to an irrational base b satisfying b^2=b+1

So for a 2 \leftarrow 1 machine we see that the base is 2 because 1b=2 :

On the other hand a 2\leftarrow 4 machine also has base 2 because 2b=4:

 

So these two machines are representing positive integers as finite sums of “digits” times powers of 2.

What’s different is what counts as a digit.

In the 2 \leftarrow 1 machine the digits are 0 and 1, while in the 2\leftarrow 4 machine the (allowable) digits are 0, 1, 2, and 3.

This gives rise to correct, but distinct representations, but both “base 2”, (!!) coming from the ways in which the two different machines operate:

Base 10 Base 2/1 Base 4/2
1 1 1
2 10 2
3 11 3
4 100 20
5 101 21
6 110 22
7 111 23
8 1000 200

The connections between the pattern that defines an exploding dots machine, the digits that are allowable, and the algebraic number that results from the pattern, is not entirely clear.

And what happens if a pattern gives a root of a cubic as the base? Do any of the – possibly complex – roots also work equally well? And what are the allowable digits then?

For that matter, what if take the negative root of b^2=b+1 as the base for an exploding dots machine with pattern 1\vert0\vert0 \leftarrow 0\vert 1\vert 1?

http://www.crikeymath.com/wp-content/uploads/2017/09/crikey.wav

Filed Under: Uncategorized

Recent Posts

  • Does the DNA test say guilty or not guilty?
  • The Turtleback Diagram for Conditional Probability
  • Numbers that, base 2, are blindingly obviously divisible by 3
  • A mysterious sequence of 1s and 2s
  • Does one have to be a genius to do mathematics? Neither necessary nor sufficient.
  • Alexander Bogomolny
  • Binary disjoints of an integer
  • A cycle of length 5 for iterated squaring modulo a prime
  • The diagonal of the Wythoff array
  • Leonardo Bigollo, accountant, son of Bill Bonacci

Archives

Copyright © 2023 · Dynamik Website Builder on Genesis Framework · WordPress · Log in