🤔 worldbuilding number systems

• on the tails of the disappointing [[2023-08-12 frosthaven puzzle one - page 3]]
• so many more interesting options
• have "system" and "notation", which are related, but separate
• ex. rivenese
  • base 5 with box notation
    • http://calyxa.com/pearl/numbers.html

systems

• different base
  • binary, base 2, related 2^N
    • good radix economy
    • base 16 notation with https://en.m.wikipedia.org/wiki/Gray_code
      • also related to Towers of Hanoi and Chinese Rings Puzzles and Tesseracts
      • rotary encoding: https://en.m.wikipedia.org/wiki/Rotaryencoder#Grayencoding
      • nicely cyclical
    • could combine with compression, ex. run-length-encoding
    • could combine with error checking, ex. hamming codes
  • base 3
    • best integer radix economy
    • ternary search trees - space efficient
      • tree based notation???
    • related to sierpinski triangle and cantor set
    • balanced ternary https://en.m.wikipedia.org/wiki/Balanced_ternary
      • uses 1, 0, -1 (or T)
  • base e
    • optimal radix economy
    • could have some marker for "log", to allow for integers
  • base sqr-root-2
    • https://en.wikipedia.org/wiki/Non-integerbaseofnumeration#Base%E2%88%9A2
    • conversions to base 2
    • properties related to squares, octagons
  • base phi
    • golden ratio
    • https://en.wikipedia.org/wiki/Goldenratiobase
    • has fun non-unique representation
    • can make use of negative exponents
• non-positional (ex. roman)
• mixed radix
  • factorial
    • https://en.m.wikipedia.org/wiki/Factorialnumbersystem
    • still has unique representation!
    • related to Lehmer Codes
    • some geometric properties
    • number of digits increases with position
      • triangle representation?
    • https://math.stackexchange.com/questions/3020533/geometrical-interpretation-for-the-sum-of-factorial-numbers
• combinatorial
  • https://en.m.wikipedia.org/wiki/Combinatorialnumbersystem
  • https://planetcalc.com/8592/
• recursive
  • https://11011110.github.io/blog/2021/12/19/recursive-bijective-numbering.html
  • fractal trees notation?
    • [[2023-08-13 number system notation explore.svg]]

      0         -         ()            
      
      1         0         (())          
      
      2         00         (()())               
      3         10         ((())())   
      
      4       000         (()()())   
      5       010         (()(())())        
      6       100         ((())()())
      7       110         ((())(())())
      8       200         ((()())()())
      9        210         ((()())(())())
      10      300         (((())())()())
      11        310         (((())())(())())
      
      12     0000         (()()()())
      13     0010
      14     0100
      15     0110
      16     0200
      17     0210
      18     0300
      19     0310
      ..
      107
      
      108   00000
      

• ???
  • https://en.m.wikipedia.org/wiki/Goodstein%27stheorem#Hereditarybase-n_notation

notations

• for positional / radix systems
  • can have various directions
    • (traditionally) big->small, but could have small->big, or vertical, or horizontal
  • trees
  • https://en.wikipedia.org/wiki/Bijective_numeration (ie. no zeros, unambigous)
  • https://en.m.wikipedia.org/wiki/Cistercian_numerals
  • https://en.m.wikipedia.org/wiki/Pentadic_numerals
  • https://en.m.wikipedia.org/wiki/Maya_numerals
  • https://en.m.wikipedia.org/wiki/Kaktovik_numerals
• sign-value notation
  • ex. base 10, + = 100s, - = 10s, . = 1s
    • 324 = +++–....