Hilbert's Building Blocks
Investigating space curves to construct 3D 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 3D forms. Not having
a lot of luck in getting results that could suggest reasonable 3D forms,
I moved back to some earlier work I did in 2D 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
Copyright 1995 Robert J. Krawczyk All Rights Reserved