## Hilbert's Building Blocks

Investigating space curves to construct 3-D forms

### Curve Generation:

I have been interested in the area of computer generated forms, mostly from the architectural viewpoint, for a long time. Most recently I have been investigating fractals as a way of generating 3-D forms. Not having a lot of luck in getting results that could suggest reasonable 3-D forms, I moved back to some earlier work I did in 2-D with Hilbert curves, spirolaterals, space filling curves, and recursive designs.

The image above on the left is the space filling curve designed by the German mathematician David Hilbert. The adjacent image shows the three line segment "generator" for the Hilbert curve. The generator is connected to another generator by a connecting line segment. By definition, this type of curve will always remain in a two dimensional plane.

If you break the generator into forward moves and turns, and then modify the angle of the turn, the lines segments will cross each other. This crossing enables the curve to trigger a move to another "level". This enables the determination of the curve height.

Variations can be developed by using a turning angle other than 90 degrees. Two such variations are shown below.

### Variations:

Variation 135/2  Variation 120/2

### Interpretation:

The second part of the this investigation is the interpretation of the curve once it is generated. Each of the line segments and their vertices can be interpreted in three dimensional, architectural terms:
• walls, each line segment is constructed as a vertical plane
• floors, for each set of line segments, the minimum and maximum extends are found and constructed into a horizontal plane
• floor blocks, the horizontal floor plane is constructed into a volume
• extended walls, walls are constructed from the bottom and the top, starting at their beginning level, extending either to the bottom or top
• columns, volumes are constructed at the vertices of the line segments and the floor blocks
• beams, volumes are constructed along each line segment at the wall height
Select one the above variations to view these interpretations individually and in combination.

### Future Directions:

The more I worked with these variations and their interpretations, the more sculptural the forms became, further studies will continue in both the sculptural and architectural form possibilities.

The next set of forms will use spirolaterals and more generalized recursive curves for the initial form generation.

The forms currently only exist in this digital studio. My next goal is to generate STL files of the forms to send to a rapid prototyping system.

Another possible direction would be to rewrite the generation software in AutoLisp for use within AutoCAD R13. This would also allow for the automation of the rendering of each variation.

The entire idea of generating forms from specifications, have software develop alternative interpretative forms, then going to physical models is very intriguing; these concepts will continue to be the general direction of this investigation.

### Technical Information:

A program written in MicroSoft QuickBASIC is used to generate all of the three dimensional components required for a particular variation in a 3D DXF format. The DXF file is then imported into AutoDesk 3D Studio for rendering. No manual modeling is required.

These Web pages were constructed for use with Netscape 1.1.

### Acknowledgements:

The initial programs which I wrote for the two dimensional versions and interpretations were upgraded to handle three dimensions by Amy Ferguson, a Teaching Assistant in the College of Architecture. She also did some exhaustive studies of the variations possible and some of the studies leading to the renderings produced here.

For further information or comment contact: Robert J. Krawczyk

Last update: Friday, December 29, 1995