Hey, this isn't original research--it's in a book. We can put it in Wikipedia! Warren privately complained that Minskys & Trinskys never formally defined them. This should obviously be answered by Julian or NeilB, but they have tastier fish to fry. So here is the attempt of a dyed-in-the-wool informalist. A Minsky is a polygon in R². It is completely specified by two real parameters, δ and ε, and any one of its vertex points. Each vertex precisely determines its two neighbors, and thus their neighbors, ad infinitum, via two exactly reversible skew transformations, linking the vertices in a first-order recurrence. x[n]==x[n-1] - f[δ,y[n-1]]==x[1+n]+f[δ,y[n]], y[n]==y[n-1] + g[ε,x[n]]==y[1+n] - g[ε,x[1+n]] If f and g in the skew transformations are linear, the recurrences solve in the usual way, and all the vertices lie on a fixed ellipse or hyperbola. But if f and g involve nonlinear functions, e.g. Floor, the polygon can become extremely interesting. Since every point has a unique predecessor, the polygon is either a closed loop or an infinite arc. There can be no loop with a tail. A Trinsky is like a Minsky, but with three (reversible) skews instead of two. The vertices of a linear Trinsky lie on an axis-aligned ellipse. Nonlinear Trinskys correspond 1-1 with Minskys, but are unskewed and less elongated, and thus easier to visually interpret. In the book, f[d,y] := Floor[d y] and g[e,x]:= Floor[e x], and the x,y "rugplots" were made by assigning a single color to all the x,y points (pixels) of each Minsky, according to its period. (Polygon-valued) Minsky(x。,y。, δ, ε) naturally partitions 4-space into patches where sufficiently small changes in x。,y。, δ, ε produce congruent polygons. Equicoloring the points of these patches rivals the Mandelbrot Set in intricacy. E.g., there are accumulation points of patches. There is a spot near (0,8,50/23,ε) where the Minskys are period 224 for ε > 10/7, period 202 for ε < 10/7, but period 18761994037932 for ε = 10/7. The Minsky(1,0,9/17,15/2) is probably finite, but exceeds 10^16 vertices. Instead of its period, NeilB and I recently experimented with winding number and winding angle about 0,0. As Neil predicted, this proved disappointing. When we chose δ ε = 4 (sin π/p)^2 , so that the recurrence had "theoretical" ( linear, no Floors) period p, and actual period P, the winding number w was within 1 of P/p. Slightly prettier was the pointillist effect of coloring according to P - w p. Also promising was rational p. Decades ago I experimented with deliberately inaccurate f and g, producing only |integer values| <= 2. This made Minsky orbits resembling potatoes with sides parallel to an octagon. Make a Lissajous figure, taking x from one potato and y from another, and plot the winding number mod 8 of each point in the filled a rectangle. This produces subrectangles of texture according to the cross product of the two 1D phase spaces. During the course of the LCM of the x and y periods, certain rectangles momentarily vanish. gosper.org/IMG_0343.JPG . This can be useful for purfling personal stationery. gosper.org/IMG_0344.JPG . While digging these up, I found gosper.org/IMG_0345.JPG , described by the young Marco Aiello as an "exploding surface", when all I see is a pillow. It is the result of much less severe quantization of f and g. --rwg These three hardcopies were produced on one of the supersecret very earliest laser printers. It only worked while the legs of a technician named Julian Orr protruded from beneath it. Solid colors--no-halftoning. We weren't supposed to take samples out of the building.