The Desargues Configuration

Introduction.

The Desargues configuration is a geometrical system consisting of 10 lines and 10 points arranged in such a way that 3 points lie on every line and 3 lines lie on every point. The configuration plays a prominent role in the somewhat outmoded study called "projective geometry". If an introductory course in this subject is taught properly, then one must inevitably encounter the following "Desargues theorem":

If two triangles are perspective from a point, then they are perspective from a line.

Since this is a course in projective geometry, one automatically has the dual statement, obtained by interchanging the words "point" and "line". In fact, the theorem is true for all projective spaces over the real numbers. Nevertheless, this theorem is a falsehood for some projective spaces. Naturally enough, one calls a space "Desarguesian" or "non-Desarguesian" depending on the verity of this statement. After seeing this theorem, one uses it to define harmonic sets, perspectivities, projectivities, and eventually prove the fundamental theorem of projective geometry.

As one of the references below describes, there are 8 different plane configurations which possess exactly 10 lines and exactly 10 points arranged in such a way that 3 points lie on every line and 3 lines lie on every point. Another characterization is needed: If such a configuration is given, then it is a Desargues configuration if, given any embedded complete quadrilateral, the remaining elements comprise a complete quadrangle. A "complete quadrilateral" is an any arrangement of 4 lines in general position taken with all 6 intersection points, and, dually, a "complete quadrangle" is any arrangement of 4 points in general position taken with all 6 of the lines joining all the pairs of points.

Here is another way to characterize the Desargues configuration. Notice that among the subsets of {1,2,3,4,5}, exactly 10 of them are comprised of two elements {a,b} and exactly 10 of them are comprised of 3 elements {c,d,e}. Having drawn a Desargues configuration, one may mark each point with a subset consisting of two elements and and mark each line with a subset consisting of three elements. A point {a,b} then lies on the line {c,d,e} if and only if {a,b} is a subset of {c,d,e}.

As one may expect, it is possible to depict the Desargues configuration:

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Figure 1. The Desargues Configuration, Java Sketchpad.

In this sketch, one should notice 3 black lines, 3 red lines, 3 blue lines, and 1 green line. The black lines are concurrent, and the red and blue triangles are perspective with respect to this point of concurrency. The Desargues theorem says that the two triangles are persective with respect to a line, and this is drawn as green in the sketch. One may move any of the 7 points which lie on the black lines in order to see the effect on the rest of the sketch. It is also a nice exercise to draw this manually, labelling the points and lines with subsets of {1,2,3,4,5} as described earlier

Zome Models.

So far, this has been a tremendous amount of background on the topic that you came for, so here we go. The following is a Zome model of the Desargues configuration:



Figure 2. Zome Model of the Desargues Configuration.

One should notice first that there are indeed 10 lines. Three of the lines are each represented by a single strut and the other seven are each represented by 2 collinear struts. One should thus see a total of 17 struts. Naturally, one uses connector balls to represent points of the configuration. In the photographed model, however, notice that there are only 9 balls, and so only 9 points are represented in the model. This is closely related to the fact that three of the lines are represented by only one strut. Notice that these three lines are mutually parallel. In the language of projective geometry, one says that these lines intersect at the "plane at infinity".

Desarguesian or non-Desarguesian?

One can make at least a dozen different models of the Desargues configuration using Zome, but one will notice that it is extremely difficult to make one for which all 10 points are confined to finite space. That is, it appears likely that every Zome model of the Desargues configuration has a triple of mutually parallel lines and thus a point at infinity. Such an "ideal" model should have the following properties: (a) It uses exactly 10 connectors. (b) It uses exactly 20 struts. (c) Each line is represented by 2 struts with 3 connectors serving as endpoints.

Given that it is possible to make so many models which are "nearly ideal" and that, generally, if a nice Zome model exists, then it is usually easy to make, it seems reasonable to conjecture that an ideal Zome model of the Desargues configuration does not exist. In other words, one might conjecture that Zome is "non-Desarguesian", if one broadens the meaning of this term slightly.

Addendum.

As has been described above, it is possible to make at least a small handful of Zome models of the Desargues configuration which fall just short of having the ideal property of realizing all 10 points in finite space. Indeed, one can make at least a dozen models having all of the following properties: (a) It uses exactly 9 connectors. (b) It uses exactly 17 struts. (c) For each line represented by 3 connectors and 2 struts, the 2 struts are in golden proportion. (d) The 3 parallel lines (with their intersection at infinity) are represented by 3 struts having 3 different lengths. Notice that the three mutually parallel segments must also be in golden proportion. Luckily, Zome struts are made in 3 different lengths for each color!

Notice finally that there appear to be two different species of these "9-ball" models. The following photographs show both of these species. One of the species has 1 black ball, 1 purple ball, 2 green balls, and 5 white balls, while the other has 2 purple balls, 3 green balls, and 4 white balls. The color of any particular ball designates the the number of struts emanating from that ball; white, green, purple, and black correspond respectively to 3, 4, 5, and 6.

Figure 3. Two Species of Zome Models of the Desargues Configuration.

References.

H. S. M. Coxeter. Introduction to Geometry. 2nd ed. John Wiley & Sons, New York, 1969.

Burkard Polster. A Geometrical Picture Book. Springer-Verlag Inc., New York, 1998.


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