1.6.2
Circle Numbers
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Circle numbers represent magnitude using a depth-value notation that supports a maximally factored form, as opposed to conventional Hindu-Arabic numbers that use place-value to represent magnitude in a polynomial form. Both use a uniform base; both support an arbitrary choice of base value. The factored circle form provides distinct computational advantages, at a slight cost to ease of readability. Maximally factored numbers are clumsy when expressed in conventional notation, but quite elegant when expressed in a boundary notation. More fundamentally, boundary notation shows that the foundational properties of group theory are largely syntactic.

boundary math
numerics

number systems
∆ circle numbers
graph numbers

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Conventional polynomial and factored boundary forms of binary twelve:

Three animations follow. They each display the same materials, but with varying degrees of condensation.

CIRCLE NUMBERS ANIMATION (complete version, 11 minutes)

CIRCLE NUMBERS ANIMATION (condensed version, 4 minutes)

ADDITION AND MULTIPLICATION OF CIRCLE NUMBERS (short version, 2:20 minutes)