Is there anything fundamental in geometry about this set of 6 transformations: Reflection, Shear, Rotation, Dilation, Squeeze and Translation?
I am looking for the cognitive or metaphysical foundations for geometry. I am especially interested in the ways we think mathematically in our imagination, implicitly, rather than the axiomatic systems that we define to constrain our work, explicitly.
I have a hypothesis that in our minds we make use of four geometries and six transformations between them. I will be presenting a related art project, a transparent doll house, at the Klaipeda Science and Art Festival on November 16. Which is to say, these are my explorations, and I am looking for connections, if any, with what is known in math.
I postulated this hypothesis after systematizing 24 ways of figuring things out in mathematics, based on a survey of methods described by George Polya, Paul Zeitz and others, as I overview in my talk, Discovery in Mathematics: A System of Deep Structure. There I imagine how four geometries (affine, projective, conformal, symplectic) could be related to how our minds generate (from the concepts of "center" and "totality") four infinite families of polytopes (simplexes, cross polytopes, cubes, demicubes) whose symmetry groups are also the Weyl groups for the root systems of the classical Lie algebras.
In math, the affine, projective, conformal and symplectic geometries have very specific meanings and so I will rename them below, respectively, as I understand them intuitively. I imagine our minds apply them to organize our expectations and evoke corresponding moods, as I note in my talk, A Research Program for a Taxonomy of Moods. And then I will be able to point out the 6 transformations which my question is about.
Vector geometry (affine): What can be constructed and deduced from one-directional vectors. For example, in my talk on moods, I consider the conditional sadness evoked by Li Bai's poem "Quiet Night Thoughts" in that beyond his bed, in the beauty of the moon and the surprise of frost-like ground is also his happy home.
Line geometry (projective): What can be constructed and deduced from two-directional lines. Suppose that the poet can look back and forth at themselves in time or otherwise.
Coordinate geometry (conformal): What can be constructed and deduced from an orthogonal basis. We can consider how people's expectations and moods are "perpendicular" or "parallel" or are given by some angle.
Sweep geometry (symplectic): What can be constructed and deduced from sweeping out an area (or volume etc.) by holding one dimension fixed while varying another. I am trying to imagine area in a dynamic sense, as I suppose happens in multiplying Position x Momentum. In my talk, I discuss the Beatles' song "She Loves You", where one person's mood is fixed while another person's mood changes.
In reading about geometry, it seems to me that, in practice, such different mindsets are not kept separate. For example, I appreciate very much Norman Wildberger's videos on Universal Hyperbolic Geometry, but I imagine that his very use of algebraic coordinates means that, in actuality, his geometric approach is not affine or projective but conformal. That is fine mathematically, but it may obscure what we do cognitively. Similarly, I think that Geometric algebra, Clifford algebra and visual complex analysis are all by nature symplectic. I am simply trying to tease apart the layers of geometry, cognitively.
At this page at Sylvain Poirier's website, I found a list of 6 transformations. I am wondering if they are, in any sense, fundamental. I interpret them below as contributing the precision needed for a geometry that is more vague to be understood as more specific.
- Reflection takes us from Vector geometry to Line geometry. A vector (and the ray it builds) can be flipped back and forth within a two-directional line, that is, the vector is given a precise orientation.
- Shear mapping takes us from Vector geometry to Coordinate geometry. A vague parallelogram can be made precise as a rectangle.
- Rotation around an origin takes us from Line geometry to Coordinate geometry. A location on a line through the origin can be given precisely as projections onto a coordinate grid, thus identifying the line with a rotation.
- Dilation takes us from Coordinate geometry to Sweep geometry. An angular shape is sized as needed so that it has a specific area.
- Squeeze mapping takes us from Line geometry to Sweep geometry. It balances the contributions that different axes will make to an overall area.
- Translation takes us from Vector geometry to Sweep geometry. It sweeps a vector (discretely or continuously) to define a dynamic area (inheriting a well defined geometry).
To be as clear as I can, I will describe these transformations algebraically, although, as I mentioned above, that imposes a coordinate system which I think is not present in the Vector and Line geometries as I understand them.
- Reflection: $(x)\rightarrow(-x)$
- Shear: $(x,y)\rightarrow(x + ay, y)$
- Rotation: $(x,y)\rightarrow(x\cos(a) − y\sin(a), x\sin(a) + y\cos(a))$
- Dilation: $(x,y)\rightarrow(ax, ay)$
- Squeeze: $(x,y)\rightarrow(x/a, ay)$
- Translation: $(x)\rightarrow(x+a)$
I was encouraged to find such well known transformations which I think play the roles that I was looking for. I think they let an object from a less specified geometry be placed within a more specified geometry. They can also help me clarify what I mean by the four geometries.
I am curious if there is any reason, mathematically, to suppose that this is, in some sense, a complete collection or not. For example, any structure of symmetries. Are there any basic transformations which I'm not including?
I wonder if this particular set of transformations, or a related set, appears in mathematics.
I am also wondering, very speculatively, what mathematical functions they might be related to. For example, I think of Alexander Grothendieck's six operations, which I hope to understand some day. I also think of six natural bases of the symmetric functions (elementary, homogeneous, power, monomial, Schur, forgotten) which I wrote my Ph.D. thesis on.
I am somewhat familiar with Klein's Erlangen program and some of John Baez's writings.
Thank you for considering my question.