AECbytes Tips and Tricks Issue
#25 (December 13, 2007)
Designing with the Golden Rectangle in AutoCAD
Book & Video Author
The golden rectangle is an ancient and sacred diagram, rediscovered in the Renaissance, encoding an advanced knowledge of natural proportion that's been used by many architects, artists, and mathematicians to design things of great aesthetic beauty. The golden rectangle is based on a simple, yet incredibly profound relationship. This relationship has called by many names over the centuries—some of which include the golden ratio, golden section, golden mean, extreme and mean ratio, and even divine proportion.
If you take any line and cut it, there is exactly one place that exhibits the following proportional relationship:
From the patterning of sunflowers, pineapples, and pinecones, to the spiral of the Nautilus shell, to the rate of multiplication of breeding pairs of rabbits, to the shape of the Great Pyramid of Giza, to the facade of the Parthenon, to Le Corbusier's Villa Stein, to the principal elevation of the UN Headquarters building in NYC, to the relationship of the length of the bones in the human hand—the golden ratio (and its Fibonacci number relatives) is sublime and everywhere.
In the golden rectangle, the length of the side of a larger square to the next smallest square is in the golden ratio. This subdivision could go on forever in ever smaller squares, but is limited to 7 squares in this piece of my personal artwork.
Google SketchUp has the "Golden Section" built-in to its drawing tools (try drawing a rectangle and you can snap to it), but AutoCAD users don't have it as easy designing with this proportion. While it's true that AutoCAD's QuickCalc has the golden ratio saved as a sample variable—which mathematicians symbolize as the Greek letter Phi—this feature is nice for calculations but not particularly useful for design.
I will show you how to construct a golden rectangle that you can use in AutoCAD, from the current version of AutoCAD Architecture 2008 back to R14. (However, dynamic block capabilities are only possible from AutoCAD 2006 onwards). This dynamic block will be interactive—you can flip it over in both directions, stretch it along its length and width, rotate it and/or move it with convenient built-in custom grips to fit any geometric situation.
Deriving the Golden Rectangle Geometrically
Your first task is to draw a rectangle with edge length of 1 unit (the unit can be inches or centimeters, it doesn't matter as we are dealing here with geometric proportion, not absolute measurement). Type POLYGON Enter 4 Enter E Enter. Toggle POLAR mode on and click the first and second edge endpoints along a horizontal line.
Next, you'll draw a circle in a special place that actually encodes the golden ratio. Turn on Endpoint, Midpoint, and Quadrant running object snaps. Type CIRCLE Enter and click the center point at the midpoint of the square's lower edge (lower red arrow below). Then click one of the square's upper corner endpoints (upper red arrow shows right corner) to set the circle's radius.
Draw in the golden rectangle by typing RECTANGLE Enter. Click the first corner point at the intersection of the circle and the square's upper right corner (upper red arrow below). Click the other corner point at the right quadrant of the circle. Select the circle and press the Delete key.
You now have a golden rectangle with one square inside. This diagram will be far more useful if you put in several additional squares. Type POLYGON Enter 4 Enter E Enter. Click the two red endpoints shown below—but click the right corner first and then the left so that the new square appears within the rectangle.
Continue using the POLYGON command four more times to add ever smaller squares. Then draw the spiral by typing ARC Enter C Enter. Click each arc's center point at the points indicated by the red arrows below. Arcs are created counterclockwise by default so remember to start and end each arc in appropriate order at diagonally opposite corners of the square in question.
Authoring the Golden Rectangle Dynamic Block
Congratulations on drafting the golden rectangle and its beautiful spiral. The next stage is to create a dynamic block that will turn this diagram into an infinitely more useful design tool.
Select everything you've drawn so far by making a window selection around it. Type BMAKE Enter and the Block Definition dialog box will appear. Type GoldenRetangle as the block name. Click the Pick Point button in the Base Point group. Pick the lower left corner of the golden rectangle diagram as the base point for this block definition. Select the Convert to block radio button and click OK. If you using a pre-2006 flavor of AutoCAD, then you're done here. Otherwise continue building intelligence into this diagram as a dynamic block.
To add dynamic features to this block you will have to take it into the block authoring environment—select the block and type BEDIT Enter. Click OK in the Edit Block Definition dialog as the correct block has already been pre-selected. The screen turns beige and the Block Authoring toolbar and palettes appear.
Dynamic blocks are made up of three parts: geometry, parameters, and actions. You've already taken care of the geometry, and now you'll create a user interface for the dynamic block by adding a series of parameters. Later you'll drive the parameters with actions.
Click the Point Parameter tool in the Block Authoring palettes and then pick the parameter location in the lower left corner of the golden rectangle (where you chose the base point of the block definition). Then click a point off to the side somewhere to set the label location.
Click the Linear Parameter tool and then pick the two left corners of the golden rectangle as start and endpoints. Click a third point off to the left side to set the label location—a process much like linear dimensioning.
Select the new linear parameter object and open the Properties palette by pressing Ctrl+1. Change the Distance label parameter to Width. In the Misc category on the Properties palette, set Number of Grips to 1. You are doing this so that the block will scale from its base point.
In much the same fashion, add another linear parameter called Length and set its number of grips to 1. Refer to the diagram below to add a rotation parameter centered on the base point. Set its default rotation angle at 90 degrees and set its label somewhere along the left edge of the golden rectangle. Add two flip parameters horizontally and vertically across the golden rectangle, being careful to place them at the rectangle's midpoints. Finally, add a base point parameter to the lower left corner of the golden rectangle.
Actions are what drive dynamic blocks. Switch to the Actions palette and start wiring up your dynamic block by associating one action with each parameter. The linear parameters get Scale actions. The angular parameter gets a Rotate action, the flip parameters get Flip actions, and the position parameter gets a Move action. In every case type ALL Enter when prompted to select objects affected by each acton. It is a good idea to place each action close to the parameter it drives.
When you've finished designing the dynamic block, click the Close Block Editor button in the authoring environment's toolbar. Select the dynamic block in the drawing window and you will see its custom grips. Click each grip and test it. The arrow grips mirror the block so you can transpose the base point to each corner of the golden rectangle by flipping one way and then the other. The round grip allows you to turn the block about its base point. The triangular grips allow you to instantly resize the golden rectangle along its length or width. The square grip moves the whole block.
If anything doesn't work as advertised, go back into the block authoring environment and fix it, or download a working copy here.
Have fun using this dynamic block and integrating the divine proportion into your designs. The final image shows a residential project I developed for my video product on ADT 2006, which made heavy use of the golden rectangle in plan and section.
This brings me to the end of this tutorial—I
hope you enjoy working with proportion and sacred geometry as much as I do. A video showing this technique
is available as Episode #3 in my video podcast, The
About the Author
Scott Onstott is a book and video author of
AEC software tutorials. He has a degree in architecture
from UC Berkeley and has served as an instructor
there, in addition to working in several prominent
engineering, architecture, and interiors firms
in San Francisco. He has also worked as a technical
editor and technology consultant.
Scott has contributed to over two dozen books
and videos on AutoCAD, Architectural Desktop,
VIZ Render, Revit, 3ds Max, VIZ, Photoshop, Illustrator,
Painter, Fireworks, and Dreamweaver. He most recently
with Lynn Allen. He can be reached via: www.ScottOnstott.com.
If you found this article useful and have not
yet subscribed to AECbytes, please consider
doing so. Subscription
is free, and more subscribers will allow this
publication to provide more of such content
Tricks > Designing with the Golden Rectangle in AutoCAD >