Mathematical visualisation

Mathematical microcosms are created by expressing mathematical structures or processes as multimodal objects. Not surprisingly, the resultant systems reveal structures not explicitly encoded in the mapping. This is not unlike deducing relationships from axioms; the choice of mapping creates (visual) primitives from which more complex structures can be constructed or recognised or 'deduced'.

Diverse mappings create diverse microcosms. Interrelationships between these microcosms are necessarily tight, either being explicitly defined in the multimodal mappings themselves, for example the same parameter to colour vs space vs time, or being implicitly induced across the multiple representations of the 'same' mathematics, for example between a local and global view, or a minimum dimension vs higher dimension phase space.

 

 

 

 

 

Notations and multimodal formalisms are required to read the space. To some extent they evolve with it and are not independent of its content. But there is also a generic element; just as reading mathematical equations is taught, reading and moving between different levels of the mappings becomes the modus operandi of the space.

The example considered is rational numbers mod1, or related structures, with a focus on the interplay between the algebraic and geometric viewpoints. The embedding of the objects in a continuous geometric space (for viewing) does not prohibit thinking of them as discrete icons or symbols standing for themselves. Similarly static representations extended in space can be thought of as representing objects flowing in time, and visa versa. Tracking these shifts in perceptualisation ultimately requires some form of mapping. This is not unlike navigating and making sense of diverse game environments without explicit instructions.

See Abstraction and Dynamic Simulation (2005)

In the cube above rational grids of primes 5, 11, 17 are encoded in green(s), primes 7, 13, 19 are encoded in blue(s), and the composite 12 is encoded in red.

The three axes of the cube can be rotated or flipped by holding down the mouse on the object and using the keys 'e', 'r', 't' for rotation, ('e' is the identity), and 'y', 'u' and 'i' for swapping or flipping pairs of axes. Holding down the mouse AND one of these keys can create an object in a cycle of continual rotation or flipping of axes, however actual behaviour seems to depend on your system and how many other programs or browser tabs you are running simultaneously!