Math cast into fantasy:

DeepLand is an alteration of the Sphereland fantasy in which focus turns away from creatures and things and shifts to properties and behavior of the atoms they are made of. Their universe is a metric-space of ‘deep’ atoms that grow from a common center. The sphere vanishes from the universe and becomes merely a mathematical boundary

In the manner of Edwin Abbott’s Flatland story in 1884, and Dionys Burger’s Sphereland sequel in 1965, the Toy-Physics story describes a rigorously geometric and dynamic mathematical metric-space (relation-space) as if it is an alien universe that is parallel to ours - a ‘toy’ universe, called DeepLand, with properties and behavior in astonishing alignment with observations in our own universe.

But unlike Flatland and Sphereland, this story is not about shapes or social order, it is about the nature of atoms with regard to energy, motion, and collective and relativistic behavior. It is a visual story made rigorous through derived elements that are not to be found in contemporary textbooks or teaching materials. It was a great struggle to get to the beginning of the story by thinking differently about the Sphereland idea of creatures living in the surface membrane of an expanding sphere. This picture depicts a breakthrough idea about the nature of a Sphereland atom, called a relational atom (r-atom).

From within the surface of the spherical boundary, an atom would appear to be a circular disc, but this mathematics begins by imagining it to be an opening into the atom’s radial spinal body beneath the surface. Expansion causes primal substance to enter the spinal body. Sensed merely on the surface, the circular disc would seem to be a cauldron of behavior on the inside (from the inrush of substance). This aligns with our universe (one dimension higher) where an atom appears to be a tiny sphere with a cauldron of behavior on the inside.

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