There are three main results in this dissertation.
The first result is the construction of an abstract visual space for rational
numbers mod1, based on the visual primitives, colour, and rational radial
direction. Mathematics is performed in this visual notation by defining
increasingly refined visual objects from these primitives. In particular,
the existence of the Farey tree enumeration of rational numbers mod1
is identified in the texture of a two-dimensional animation.
¶
The second result is a new enumeration of the rational numbers mod1,
obtained, and expressed, in abstract visual space, as the visual object
coset waves of coset fans on the torus. Its geometry is shown to encode
a countably infinite tree structure, whose branches are cosets, nZ+m,
where n, m (and k) are integers. These cosets are in geometrical 1-1
correspondence with sequences kn+m, (of denominators) of rational
numbers, and with visual subobjects of the torus called coset fans.
¶
The third result is an enumeration in time of the visual hierarchy of the
discrete buds of the Mandelbrot boundary by coset waves of coset fans.
It is constructed by embedding the circular Farey tree geometrically into
the empty internal region of the Mandelbrot set. In particular, coset fans
attached to points of the (internal) binary tree index countably infinite
sequences of buds on the (external) Mandelbrot boundary.
| Identifer | oai:union.ndltd.org:ADTP/216719 |
| Date | January 2000 |
| Creators | Tolmie, Julie, julie.tolmie@techbc.ca |
| Publisher | The Australian National University. School of Mathematical Sciences |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | http://www.anu.edu.au/legal/copyright/copyrit.html), Copyright Julie Tolmie |
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