chriswarbo-net: 8058caed835f700728bab8d58ef267f404772d6c

     1: ---
     2: title: Extents (work in progress)
     3: ---
     4: 
     5: <small>Part of [my units pages](/projects/units)</small>
     6: 
     7: TODO: This is closely related to the idea of [Torsors](torsors.html), and the
     8: difference between affine points being vectors.
     9: 
    10:  - This is the idea of a bounded *region*
    11:  - It can be defined by its boundary, e.g. start and end points in 1D
    12:  - It is distinct from an *interval* in two ways:
    13:     - It is *relative*, e.g. more like a 1D vector
    14:     - It is *signed*, again more like a 1D vector
    15:  - Useful to distinguish position (e.g. on a number line) from
    16:    difference-in-position
    17:  - Good precursr for vectors:
    18:     - In 1D, vectors *are* extents
    19:     - In higher-dimensions, vectors also have a *direction* (maybe relatable to
    20:       projective points-at-infinity)
    21:  - Higher-dimensional extents include signed area, signed volume, etc.
    22:  - In the 1D case it gives an unambiguous notion of subtraction for positive
    23:    spaces like Nat: the difference between two Nats is an extent, not a Nat
    24:     - This removes the difficulty of e.g. 1 - 2 = -2: the sign of the result
    25:       tells us which direction to move, not 'where we end up'
    26:     - We still get problems if we try asking which Nat is some extent away from
    27:       another, e.g. taking the difference between 10 and 3 (-7) and asking what
    28:       is an equivalent difference away from 2 (there is no -5 in Nat); yet this
    29:       feels like a clearer case of 'not making sense'
    30: 
    31: Numbers and geometry are foundational to mathematics, and how we explain and
    32: understand various phenomena. We can relate these concepts in two important, but
    33: distinct, ways: as *positions* or as *extents*.
    34: 
    35: ## Position versus extent ##
    36: 
    37: Consider the useful picture of a *number line*:
    38: 
    39: ```
    40: ┌──┬──┬──┬──┬──┬──┬─⋯
    41: 0  1  2  3  4  5  6
    42: ```
    43: 
    44: This shows numbers as *positions* quite directly: the number four 'is' the
    45: position (or *point*) labelled `4`.
    46: 
    47: The *extents* in this picture are a bit more abstract: we can find them by
    48: 'cutting' the line at the relevant label; e.g. cutting at the label `4` gives
    49: the following:
    50: 
    51: ```
    52: ┌──┬──┬──┬──┐
    53: 0  1  2  3  4
    54: ```
    55: 
    56: Whilst this is a perfectly good *length*, the idea of 'extent' that I'm after
    57: needs two more things...
    58: 
    59: ## Extents are relative ##
    60: 
    61: We will consider *the line* to be our extent, not the points or labels. Hence
    62: all of the following are *the same* extent:
    63: 
    64: ```
    65: ┌──┬──┬──┬──┐
    66: 0  1  2  3  4
    67: 
    68: ┌──┬──┬──┬──┐
    69: 1  2  3  4  5
    70: 
    71: ┌──┬──┬──┬──┐
    72: 32 33 34 35 36
    73: ```
    74: 
    75: Hence we can drop the labels, to get a line like `┌──┬──┬──┬──┐`