Haresh Lalvani is a morphologist working in New York City. For decades he studied two levels of the behavior of space.
One level he has mastered. Together with Michael Burt, he is our top theorist of how space is internally structured when seen in terms of repeating units. These are systems of packing where there are no holes. I believe he invented a second level which is a way of categorizing these system using the same vocabulary.
Let me repeat that. It is relatively easy to write mathematical statements that characterize how space can be divided. (Some of us would say: ordered.) It is equally easy to write a different set of equations that relates the various systems and shows how some can transform into others. This is not what Lalvani has done.
He has instead graphically shown the space-filling packings. Some were known and others discovered by him and others during the study. He also devised laws that relate these and used the same vocabulary of space and many of the same symmetries.
Said yet another way: most geometers look at classes of symmetries based on constraints like dimension, whether they tile (fill space) and so on. Others try to find overall governing laws over these gadgets and their symmetries. What Lalvani does here and elsewhere is to speculate on those laws and express them as if they had their own geometries and symmetries. What’s intriguing is that usually when you talk about these overall laws, you have to rely on math notation. Lalvani instead expresses the laws over geometric gadgets geometrically.He seems to overtly eschew math notation to convey the order of things visually.
His early work focused on understanding the order among regular polyhedra by studying continuous transformations (by certain rules) among them. For decades, he expanded this, most interestingly looking at the order of transformations among space-filling structures. The 1989 paper deals with something in between, polyhedra that can be substituted for the vertices of a seed polyhedron.
The structures described in this paper can be space filling in a sense, but it is harder to see how we can use them. The abstract with an example metastructure in six dimensions is included here. (This note will change and grow to include many Lalvani papers, so do not trust the permalink. The rationale behind this is here.)
The Relationship to Kutachi
The attraction for Kutachi is obvious, because it promises to reveal the logic of form as form.
More specifically, we need to represent large numbers of individual facts — for instance ontologies — and manipulate them
My impression is that his metastructures over regular space filling structures is directly leverageable for ordered Kutachi visualizations of ontology graphs.
Links
The Meta-Morphology of Polyhedral Clusters. Haresh Lalvani. [Symmetry of Structure] 1989.
The Kutachi essay on Semantic Distance. (not online yet)
The Kutachi essay on Virtual Environments. (not online yet)