Species operate in space and their behavioral success depends on how well they process, represent, store, and recall spatial information. Here, we review research related to inherent capabilities and limitations of brain plasticity in terms of its spatial representations and discuss whether with appropriate training, humans can build perceptual and sensorimotor representations of spatial 4D environments, and how the presence or lack of ability of a solid and direct 4D representation can reveal underlying neural representations of space. A fundamental question is whether our brains are inherently limited to 3D representations of the environment because we are living in a 3D world, or alternatively, our brains may have the inherent capability and plasticity of representing arbitrary dimensions however, 3D representations emerge from the fact that our development and learning take place in a 3D world. This adaptation continues in adulthood and is quite general to successfully deal with joint-space changes (longer arms due to growth), skull and eye size changes (and still being able of accurate eye movements), etc. Three-dimensional perceptual and sensorimotor capabilities emerge during development: the physiology of the growing baby changes hence necessitating an ongoing re-adaptation of the mapping between 3D sensory representations and the motor coordinates. These 3D representations underlie our 3D perceptions of the world and are mapped into our motor systems to generate accurate sensorimotor behaviors. By using multiple cues, such as disparity, motion parallax, perspective, our brains can construct 3D representations of the world from the 2D projections on our retinas. We use this theorem to give, for all sets E that are analytic, i.e., Σ¹₁, a tight bound on the packing dimension of the hyperspace of E in terms of the packing dimension of E itself.We live in a three-dimensional (3D) spatial world however, our retinas receive a pair of 2D projections of the 3D environment. (For a concrete computational example, the stages E₀, E₁, E₂, … used to construct a self-similar fractal E in the plane are elements of the hyperspace of the plane, and they converge to E in the hyperspace.) Our third main result, proven via our extended point-to-set principle, states that, under a wide variety of gauge families, the classical packing dimension agrees with the classical upper Minkowski dimension on all hyperspaces of compact sets. We demonstrate the power of our extended point-to-set principle by using it to prove new theorems about classical fractal dimensions in hyperspaces. Our first two main results then extend the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families.
We first extend two fractal dimensions - computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions dim(x) and Dim(x) to individual points x ∈ X - to arbitrary separable metric spaces and to arbitrary gauge families. In this paper we extend the reach of the point-to-set principle from Euclidean spaces to arbitrary separable metric spaces X.
These are classical questions, meaning that their statements do not involve computation or related aspects of logic. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces ℝⁿ. Go to the corresponding LIPIcs Volume PortalĮxtending the Reach of the Point-To-Set Principle
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