Regular Polyhedral Lattices of Genus 2: 11 Platonic Equivalents?

The paper observes Euler's formula for genus 2 regular polyhedral lattices is obeyed by at most 11 cases of the Schläfli symbol {p,q}, where p is the number of edges of each face and q the number of faces meeting at each vertex. At least one example is given for the 'first' 6 cases, but not for their 5 'duals'. The examples are known from various sources, but their present classification suggests they are lookalikes of classical Platonic equivalents. An 'artistic' corollary is the observation that hyperbolic geometry models can be constructed using Zometool

Reverse fishbone perspective

After the discovery of perspective during the Renaissance, the rules of perspective became so familiar people began to look down at earlier painters or artists from other cultures, who did not follow those rules, such as the ‘Flemish Primitives’, or Byzantine iconographic drawers. In this paper we recall a method for explaining the well-known regular perspective and parallel projection based on the classical Monge top view and profile view. Next, we combine regular perspective and parallel projection to get Panofsky’s so-called ‘fishbone perspective’, showing it is the logical result of an algorithmic construction. It also illustrates the analogy between the vertical fishbone method and the horizontal one. It can be combined with the above mentioned reverse perspective in which objects are drawn as if they are seen from some imaginary point behind the screen and above the observer, that is, from ‘the heavens’ (though the present paper argues it should rather be from a point below, from ‘hell’). The algorithmic construction methods also explain why there are intermediate forms, and thus the critiques, using mainly philosophical arguments, were perhaps too unforgiving.status: publishe