holla bots,

Christopher Carlson in his Wolfram Blog (+) provides an uber cool mathematical run-through of your favooorite projects. hah

“It started with an innocent experiment in lofting, a technique also known as “skinning” that originated in boat-building. I wanted to explore some three-dimensional forms, and a basic lofting function seemed like a quick ticket to results. I dashed off the functionLoft, which takes a stack of three-dimensional contours and covers it with a skin of polygons.”

"Loft uses Mathematica’s GraphicsComplex primitive to factor out the geometries of the polygons from their topologies. The contour point coordinates are collected in the first argument. The second argument is a list of Polygons whose coordinate values are replaced by integer indices into the coordinate list. My Loft function was straightforward to write, but required a little fancy footwork with indexing to get the polygons wired onto the points in the right way.”

“Even this trivial parameterization of a scaled and twisted half-sphere yields an amazing variety of  forms, each of which suggests interesting avenues to explore.”

The last of those forms brought to mind Norman Foster’s Swiss Re building in London, nicknamed by the locals “the Gherkin.”

“I wondered how convincingly I could model the Gherkin in Mathematica. It was immediately obvious that my simple Loft function was not up to the task of replicating the white diagrid framing structure employed in the Gherkin, so I set out first to generalize Loft. One thing lead to another, and soon I had the much larger but much more flexible function Build, with which I could explore not only Foster’s Gherkin but a large number of other architectural forms based on the simple idea of hanging panes, panels, mullions, and framing members on grids of points.

My Build function works like Loft, but gives me much more flexibility in specifying elements like tubes and polygons and how they are repeated on the contour grid. LikeLoft, Build’s first argument is a set of contours. The second argument is aGraphics3D-style primitive list whose primitives contain an extra argument that specifies how they should be repeated on the contour grid.

If you imagine the contours numbered from bottom to top and the points in the contours numbered from left to right, {point, contour} indices correspond to coordinates in an integer coordinate system. The primitive
Polygon[{{0,0},{1,2},{1,0}}] appears on a contour grid like this.”

“In linear primitives like Line and Tube, the repetition argument specifies the frequency with which the primitive is repeated horizontally, or for horizontal primitives, vertically. By combining repetitions of polygons, tubes, and lines, Build gives me great flexibility in describing assemblages of panes and structural members. Here’s an abstract structure I generated to exercise all of Build’s primitives.”

As a final step, I refined the contour points at the top to add the dome-like cap at the top of the Gherkin.

I won’t deny that from there it required a surprising amount of detailed work usingBuild to make a finished model. The Gherkin’s body, its cap, the topmost dome, the rings, and the boundaries of and transitions between the separate parts all required individual attention. To select material properties and lighting, I set up Manipulaterigs and exercised the sliders until I found the right values. Here is the result.

Post-processing via replacement rules can operate on the geometry of an object as well as its appearance attributes. Because all of the coordinate data of my model resides in the first argument of GraphicsComplex, coordinate transformations are particularly easy. And since all of the graphics primitives are all wired to the same coordinates, the primitives automatically move in concert and remain connected when the coordinates are transformed.

Using that technique, I wrote this Manipulate to explore variations in the radial geometry of the Gherkin.

*BONUS*