# Hartmut F. W. Höft

When I encounter artistic expressions such as poetry, I not only attempt to find formulas for their combinatorial structure, but also abstract the essence of the written words into geometrical shapes, and visualize these shapes in two or three dimensions. With my images I hope to show the tension between strictly adhering to symmetry and the structural rules of poems on the one hand and breaking them to achieve an expressive or emotional outcome on the other. The first two images are based on the rhymes in cycles of sonnets; I transform their rhyme patterns into colored rectangular bands and unroll the patterns along a curve or project them onto a surface. The third image shows the structure of this surface through a portion of its mesh.

The rhyme patterns of the sonnets in "Die Sonnette an Orpheus" by Rainer Maria Rilke determine the coloring schema in this image. I made the band 28 strips wide by mirroring the rhyme pattern of the 14-line sonnets. Therefore, the color of the first rhyme sound is at both edges of the surface. I chose the trefoil for this visualization to express the entanglement of the three characters - Orpheus, Euridice & Hades - in the Greek myth to which Rilke alludes. The trefoil curve is defined by the parametric equation 2/3 * ( sin(3t), sin(t) + 2*sin(2t), cos(t) - 2*cos(2t) ) causing self-intersections when the rhyme pattern of each mirrored sonnet is drawn in the tangent-normal plane to the trefoil curve at the point where the sonnet is placed.

Take a string and wind it three times around a cylinder from top to bottom and back, and you trace the curve ( cos(3t), sin(3t), sin(t) ). Now modify the closed curve componentwise with the parabola (cos(s), cos(2s), 1). I wrap the rectangular colored bands of rhyme patterns for the sonnets in "Die Sonnette an Orpheus" by Rainer Maria Rilke three time around the resulting self-intersecting surface. This yields three layers of 3-dimensional quadrangles on the surface that do not quite align due to small computational differences. Therefore, the graphics rendering creates quadrangles with mixed color patterns as well as a single color; rendering only parts of the surface exposes features such as straight geodesics.

This image shows portions of two curve patterns - parabolas and closed spirals with winding number three - on a single surface. One is a collection of entire spirals at discrete values of the parabolic parameter. The other is a collection of parabolas that are attached at fixed parameter intervals along the spiral; each parabola lies in the horizontal plane of the point of attachment on the spiral. Though not immediately obvious from the equation of this surface, f(s, t) = ( cos(3t) cos(s), sin(3t) cos(2s), sin(t) ), the image shows degeneracy into points as well as self-intersection of the surface. In my imagination the surface curves in this image evoke tension spindels suspending a central band, all protected by a swirling spiral wall.