In reply to @gm_z:matrix.org
yes

I apologize if this belongs in the "Applied" room, instead, but it is tangential (Ha!) to algebraic geometry. My aim is to mine the room for better/more articulations of how to model my particular application.

I'm trying to discover some satisfying algebraic operations for computer graphics. For example, duplicating some "stuff" (elements, controls, renderable data) across the screen could be done by tiliing the screen, as in something like (R^2/\Lambda) \times s where \Lambda is a pair of points (think "lattice") and s is a stuff, which may itself be a sum of other stuffs, where a sum of stuffs is "free" in the sense that it's basically just a commutative grouping of stuffs with no concise syntactic reduction.

That example suggests that multipling a stuff s by a point p \in R^2 is just a translation of s, which seems intuitive enough. Consequently, a weighted sum of stuffs is just a new stuff with all the summand stuffs translated, as in \Sigma_i p_i s_i.

However, translation of a point p by a point p' should be a sum p' + p. So if we allow stuffs to distribute over points, we run into trouble: (p' + p)s = p' s + ps. But distributing points over stuffs works just fine: (p' + p)(s_1 + s_2) = (p' + p)s_1 + (p' + p)s_2. So this smells like a left group action of the additive group of points on the set of stuffs.

Of course, the above observations are barely a sketch; I just scribbled them down before bed, last night. But if anyone is willing to throw some random insights my way, it would be greatly appreciated! I'm just hoping to spitball enough to work out a basic implementation that would be suitable for rendering a decent graphical user interface. If a lot of needlessly fiddly little computations could be boiled down to a conceptually coherent algebraic representation, I suspect the benefits would be substantial.

I apologize if this belongs in the "Applied" room, instead, but it is tangential (Ha!) to algebraic geometry. My aim is to mine the room for better/more articulations of how to model my particular application.

I'm trying to discover some satisfying algebraic operations for computer graphics. For example, duplicating some "stuff" (elements, controls, renderable data) across the screen could be done by tiling the screen, as in something like (R^2/\Lambda) \times s where \Lambda is a pair of points (think "lattice") and s is a stuff, which may itself be a sum of other stuffs, where a sum of stuffs is "free" in the sense that it's basically just a commutative grouping of stuffs with no concise syntactic reduction.

That example suggests that multipling a stuff s by a point p \in R^2 is just a translation of s, which seems intuitive enough. Consequently, a weighted sum of stuffs is just a new stuff with all the summand stuffs translated, as in \Sigma_i p_i s_i.

However, translation of a point p by a point p' should be a sum p' + p. So if we allow stuffs to distribute over points, we run into trouble: (p' + p)s = p' s + ps. But distributing points over stuffs works just fine: (p' + p)(s_1 + s_2) = (p' + p)s_1 + (p' + p)s_2. So this smells like a left group action of the additive group of points on the set of stuffs.

Of course, the above observations are barely a sketch; I just scribbled them down before bed, last night. But if anyone is willing to throw some random insights my way, it would be greatly appreciated! I'm just hoping to spitball enough to work out a basic implementation that would be suitable for rendering a decent graphical user interface. If a lot of needlessly fiddly little computations could be boiled down to a conceptually coherent algebraic representation, I suspect the benefits would be substantial.

I apologize if this belongs in the "Applied" room, instead, but it is tangential (Ha!) to algebraic geometry. My aim is to mine the room for better/more articulations of how to model my particular application.

I'm trying to discover some satisfying algebraic operations for computer graphics. For example, duplicating some "stuff" (elements, controls, renderable data) across the screen could be done by tiling the screen, as in something like (R^2/\Lambda) \times s where \Lambda is a pair of points (think "lattice") and s is a stuff, which may itself be a sum of other stuffs, where a sum of stuffs is "free" in the sense that it's basically just a commutative grouping of stuffs with no concise syntactic reduction.

That example suggests that multiplying a stuff s by a point p \in R^2 is just a translation of s, which seems intuitive enough. Consequently, a weighted sum of stuffs is just a new stuff with all the summand stuffs translated, as in \Sigma_i p_i s_i.

However, translation of a point p by a point p' should be a sum p' + p. So if we allow stuffs to distribute over points, we run into trouble: (p' + p)s = p' s + ps. But distributing points over stuffs works just fine: (p' + p)(s_1 + s_2) = (p' + p)s_1 + (p' + p)s_2. So this smells like a left group action of the additive group of points on the set of stuffs.

Of course, the above observations are barely a sketch; I just scribbled them down before bed, last night. But if anyone is willing to throw some random insights my way, it would be greatly appreciated! I'm just hoping to spitball enough to work out a basic implementation that would be suitable for rendering a decent graphical user interface. If a lot of needlessly fiddly little computations could be boiled down to a conceptually coherent algebraic representation, I suspect the benefits would be substantial.