Drawingon DesarguesANNALISA CRANNELL AND STEPHANIE DOUGLAS

DE S A R G U E S’S T R I A N G L E T H E O R E M: Two trianglesthat are perspective from a point are also perspective from a

line.

AAbout four years before Girard Desargues wrote theLeçon deTénèbrescontaining thepreceding theorem—a theorem that bears his name to this day—he wrote a

12-page treatise on artistic perspective [6]. Desargues’sperspective treatise was a bit of an IKEA manual: it describeshow to draw a gazebo-like object of certain proportionswithout resorting to the use of vanishing points (seeFigure 1), and Desargues did not seem to expect that layartists would apply its techniques to more general objects.The book was an explanation of an example, not anexplanation of a theory; Desargues’s art treatise does notcontain Desargues’s triangle theorem. Still, projective geom-etry has flirted with perspective art ever since.

Desargues’s more mathematical Leçon de Ténèbres is lost;the closest surviving relative comes in the final several pagesof abookbyoneofDesargues’s acolytes,Bosse [2], publisheda dozen years later, pulls together both the artistic and themathematical work of Desargues. But Bosse proof of De-sargues’s triangle theorem is more mathematical than artistic.It uses the cross-ratio, a tool found in very few artists’ reper-toires. And the accompanying diagram? If you, like mostpeople, have a hard time deciphering Bosse’s arrangement(Figure 2), you can try puzzling it out by noting that trianglesabl and DEK are perspective from both the point H and theline cfg. Or you could realize that it’s really not a very helpfuldiagram.

Bosse’s is the first in a centuries-long tradition of Desar-gues’s Theorem illustrations, few of which require as mucheffort on the part of the reader as Bosse’s. Still, if you searchthrough the projective geometry books in your library, youare likely to find a lot of pencil marks. In our own searchesthrough some 20 different volumes (see [1], [3]–[5], [7]–[10]and [13]–[29]), we found we had to do a lot of erasing if wewanted a ‘‘clean’’ version of the author’s Desargues’s trianglefigures. People who have used these books over the yearshave shaded in triangles, darkened lines, and circled impor-tant points so that they could read the diagram more easily.

The point of this article is to illustrate—literally—Desar-gues’s theorem. Many authors have shown that it is possibleto ‘‘lift’’ a planar Desargues configuration into 3-dimensions;we will instead view a planar configuration as a drawing ofobjects already in 3 dimensions. Doing so gives us a lovelyand paradoxical academic symmetry: perspective art usesgeometrical tools to portray a 3-d world on a 2-d canvaswhereas we give 2-d geometrical drawings a 3-d perspectiveinterpretation.

DefinitionsFigure 3 shows two common physical ways we might projectobjects from our 3-d world onto a canvas: the candle flameprojects a shadow of the rabbit onto the floor and wall, andthe pinhole projects an image of the thinker onto the wall. Inprojective geometry we formalize these physical notions bysaying that two triangles X ; Y � R3 are perspective from apoint O if there is a one-to-one correspondence between theverticesof these triangles so thatwhenever the vertex VX 2 Xcorresponds to the vertex VY 2 Y; the three points VX ;VY ;and O are collinear. We call O the center of the projection.We’ll have occasion later to refer to ‘‘shadow’’ perspectivities(for which a triangle and its image are on the same side of thecenter) and ‘‘pinhole’’ perspectivities (for which the triangleand its image are on opposite sides of the center). As Figure 4shows, these are not the only kinds of perspectivities.

We can say that the trianglesX andY are perspective froma line ‘ if there is a one-to-one correspondence between thethree lines (the extended edges) of these triangles so thatwhenever the line ‘X � X corresponds to the line ‘Y � Y; thelines ‘X ; ‘Y ; and ‘meet at a common point. In other words,the three points where pairs of corresponding lines intersectall lie on ‘. We call the line ‘ the axis. In Figure 5, the lines ofthe shaded triangle on the paper and their reflections in themirror are perspective from the line where the mirror meetsthe paper.

It is not true in general that twoobjects that are perspectivefrom a point are also perspective from a line (indeed, thedifference between these two kinds of perspectivities formsthe basis for some spiffy optical illusions; think of M. C.Escher’s Waterfall or the sculpture of the Penrose Triangle in

� 2012 Springer Science+Business Media, LLC, Volume 34, Number 2, 2012 7

DOI 10.1007/s00283-011-9271-y

East Perth, Australia). Desargues’s theorem and its conversetell us, however, that these two notions are equivalent fortriangles. An intuitive and rather breezy explanation of thisphenomenon is that a triangle (unlike a rabbit, a hand, or athinker) uniquely defines a plane. Two triangles perspectivefromapoint canbe interpreted as a 2-ddrawingof triangles in3-space, lying in two planes whose intersection (as in theexample of the paper and mirror of Figure 5) forms the axis.We will not prove Desargues’s theorem, but we will use theabove breezy interpretation to draw our own pictures of thetheorem.

Drawing on CoxeterThere are some remarkable commonalities among the pub-lished diagrams illustrating Desargues’s Theorem. Figure 6

gives three fairly typical examples.Withoneexception, all thediagrams we have seen illustrate a shadow-type projection;Horadam [16] bucks the trend by showing a pinhole-typeprojection.More significantly, in everyoneof the books in thecollection we examined, the diagrams illustrate a case wherethe perspectivity maps the filled-in, finite interior of one tri-angle onto the filled-in, finite interior of the other triangle.That is, there are no diagrams that look like Figure 4. As wewill see, this interior-to-interior projection is often incorrect.

For all else that follows, we will interpret and reinterpret(indeed, literally draw on) thediagram that appears inwhat isarguably the best-known book of its kind, Coxeter’s Projec-tive Geometry [4]. We use this diagram not only because of its

.........................................................................................................................................................

AU

TH

OR

S ANNALISA CRANNELL received her B.A.

from Bryn Mawr College and her Ph.D.

from Brown University. Her mathematical

interests have morphed over the years

from nonlinear PDEs to discrete dynamicalsystems to – most recently – the applica-

tions of projective geometry to perspective

art. Together with Marc Frantz, she is a

coauthor of the recent Viewpoints: Math-ematical Perspective and Fractal Geometryin Art.

Department of Mathematics

Franklin & Marshall College

Box 3003

Lancaster, PA 17604-3003

USA

e-mail: annalisa.crannell@fandm.edu

STEPHANIE DOUGLAS is a 2012 graduate of

Franklin & Marshall College, majoring in

astrophysics. She enjoys dabbling in mathe-

matics on the occasions when it involves art.

She intends to study astrophysics in graduate

school.

Department of Mathematics

Franklin & Marshall College

Box 3003

Lancaster, PA 17604-3003

USA

e-mail: stephanie.douglas@fandm.edu

Figure 1. The plate illustrating Desargues’s Perspective. Figure 2. Desargues’s theorem as illustrated by Bosse, from

[11]. We have enlarged the labels from the original diagram.

8 THE MATHEMATICAL INTELLIGENCER

popularity, but also becauseCoxeter’s drawingofDesargues’sTheorem is simple in the extreme: it shows no shading orvariation in line-weight, only ten line segments and tenpoints that are indistinguishable in style. Figure 7 shows theconfiguration, although we altered the diagram slightly byremoving Coxeter’s labels and giving six of the ten points inhis diagram our own new labels. The explanation of thesenew labels will become apparent later.

Coxeter chose as the center of his perspectivity the pointwe call A3; if we shade in the resulting triangles that followfrom this choice, we get a figure like that of Eves (Figure 6). Ifwe add a few details, we get Figure 8: an illustration of a lightshining throughanobject onawindow, creating a shadow. Inaddition to ‘‘embedding’’ each triangle in a plane, we havealso used standard perspective techniques (such as over-lapping and variable line weights) to imply that some linesare closer to the viewer of the diagram than others. In thisimage, the conclusion that each edge of the triangle mustmeet its shadow at the axis becomes visually obvious.

Let’s step back to consider why the readers of our copy ofCoxeter made so many additional pencil marks in the book.All ten line segments in the diagram look the same, and that’sbecause, in an important geometrical sense, they all are thesame. Coxeter’s Desargues configuration has the propertythat each of the ten line segments contains three points; eachof the ten points lies on three lines. So what happens if wechoose a different point to be the center? The reader canverify easily that choosing a point as center automaticallydetermines the triangles and the axis. In what follows, thesechoices and their implications become the variations on ourtheme.

The next center we chose is the one we named B3. Again,we shade the interiors of the resulting triangles. In this case,because the axis passes through the interior of the triangles,we use standard overlapping techniques to indicate thatsome parts of the triangles are close and others are far. DoesFigure 9(a) show a triangle passing through its own shadow?We think this diagram is visually understandable andbelievable if we reinterpret it slightly to say that each one ofthese triangles casts the same shadow on a given plane(Figure 9(b)), in the sameway that the rabbit of Figure 3 caststhe same shadow as a hand would. [An exercise for thereader: this coloring of Coxeter’s diagram gives us the sameconfiguration as Bosse’s diagram. Can you see that foryourself?]

But our variations on this theme get weirder at otherpoints. Both points A3 and B3 have the property that they areendpoints of their three line segments—the subscript ‘‘3’’ inthe name of each point stands for the number of line seg-ments ending at that point. In Figure 10, we choose otherpoints as center and then color the interiors of the triangles aswe did before. Our diagrams look a bit like the earlier ones,but a closer look tells us something interesting is going on.Nowwhat does the axismean? If C2 is the center,whydoes theaxis pass through one triangle but not through the other?Similarly, what is going on in the diagram whose center is D2?

Figure 3. A shadow projection and a pinhole projection.

Figure 4. We project a vertical triangle onto a horizontal

plane. The parts of the triangle above the center of projection

get sent via a pinhole projection to an infinite triangle; the

parts below the triangle get sent via a shadow projection to an

infinite trapezoid. The dividing line segment gets sent ‘‘to a

line at infinity.’’.

Figure 5. The lines on the paper and their images in the

mirror are perspective from a line (the line where the mirror

meets the paper).

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Or in the diagram whose center is E1? The center of the pro-jection F0 is contained in the interior of three segments. Thatis, F0 is like the hole of a pinhole camera, . . . but how do weunderstand the dark line of projectivity? Why is it where it is,and not (for example) vertical, between the two triangles as itis in Horadam’s diagram (Figure 6)?

There are two reasons for the visual confusion of thediagrams in Figures 10. The first reason is that our ownlocation in space (our perspective, so to speak) matters.

Let us illustrate this first concern in the case of the lastexample, for which the center is F0. When two planes

intersect, they divide three-space into four quadrants. Whenwe imagine a pinhole projection, we most easily imagineourselves in the same quadrant as the pinhole, as Fig-ure 11(a). In this figure, the axis appears to lie between thetwo triangles, which matches our intuition. But when werotate the diagram in Figure 11(a), we get Figure 11(b) . . .and this latter view matches Coxeter’s diagram. That is, usingartistic techniques to create a sense of appropriate 3-d spaceallowsus toportray thediagram in away that seems ‘‘correct.’’

The second reason for the visual confusion is more sub-stantive. It has to do with the parts of the figures we chose toshade. When two triangles are perspective from a centerO, although the vertices of one triangle project from O on tothe vertices of the other, the interior of a triangle in a Desar-gues’s diagramdoes not alwaysproject onto the interior of theother triangle, for reasons that Figure 12 illustrates. The dis-connected projection of line segments in Figure 12 is relatedto the disconnected projection of triangles from Figure 4. Aswenotedpreviously, this casewas not addressed in anyof thebooks that we examined. This is why Desargues’s theoremproperly speaks of lines, not of line segments—a distinctionthat does not seem to be apparent in most graphic depictionsof the theorem.

Let us give an example of how this ‘‘inside-out’’ projectionapplies to Coxeter’s figure, looking at the projection centeredatO = D2. Figure 13 relabels thepoints in away thatwill helpus describe overlappings.

The line segment A0OA never crosses the dark axis ‘. Thattells us that the line segment A0A lies in the same quadrant asthe center O. The segment OBB0 tells us that B is on a half-plane near the center O whereas B0 is on a far one (the

Figure 6. Desargues’s theorem, as illustrated by Eves [9], Veblen and Young [28], and Horadam [16].

Figure 7. Coxeter’s diagram of Desargues’s theorem, with

Crannell’s labels. This Desargues’s configuration has the

property that each of the ten line segments contains three

points; each of the ten points lies on three lines.

Figure 8. Coxeter’s diagram with the center of projectivity at A3, oriented (left) as in Coxeter and (right) with the axis represented

in a horizontal position. The axis ‘ is represented by the intersection of the two planes containing the respective triangles.

10 THE MATHEMATICAL INTELLIGENCER

intersection of the axis with the segment BC 0 confirms this).Likewise, C is on a near half-plane and C 0 is on a far one. Thatis, both triangles cross the axis. See Figure 14.

The last step in drawing an interpretation of the diagram isto note that the interiors of the triangles in this diagram aren’treally projective images of each other, even though the lines

(b)(a)

Figure 9. (a) Coxeter’s diagram with the center of projectivity at B3. (b) We may think of this diagram as saying that many

different triangles can cast the same shadow.

Figure 10. Coxeter’s diagram with the center of projectivity at C2, D2, E1, and F0. We fill in the interiors of the resulting triangles,

but something strange is going on here. How should we interpret these diagrams?.

(a)

(b)

Figure 11. A pinhole projection. In (a), we are in the same quadrant with the pinhole. In (b), we rotate the preceding pinhole

figure to get an interpretation of Coxeter’s diagram centered at F0.

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that form the edges of the triangles are. Instead, as withFigure 4, the projective image of the finite triangle is a com-bination of an infinite trapezoid (toward the top of Figure 15)andan infinite triangle (on thebottom right sideof thefigure).

The trapezoid is the result of a shadowprojection; the triangleis the result of a pinhole projection. If we rotate Figure 15clockwise 90 degrees, as in Figure 16, the similarity to Fig-ure 4 is all the more striking.

We may proceed in the same way to draw perspectiveviews of Coxeter’s diagram with the center at C2 or E1, as inFigures 17 and 18. For example, we might see the arrange-ment whose center is at E1 as a triangle passing through itsimage plane (as in the left of Figure 18); if we rotate thisconfiguration toward us, we will get the Coxeter diagram onthe right.

Given a projective diagram (such as Coxeter’s), it is a hard-but-worthwhile exercise to figure out how to represent thatdiagram accurately as a meta-projection. Readers who wantto see how tricky this is should cover up the figures in thisarticle and try their hands at C2 or E1! Andoften, as thepinholeexample shows, it’s helpful to have several meta-versions,

Figure 12. The center O projects the points A and B on the

line ‘ to the points A0 and B0 on the line ‘0, but it projects the

line segment AB to the complement of the line segment A0B0.

Figure 13. Coxeter’s diagram with the center of projectivity at

D2. We have labeled the points and the axis so that we can try

to figure out where in the world things are.

Figure 14. A projective image of a strange-looking projectiv-

ity. The vertices of the triangles are projective images of each

other, but the interiors of the triangles do not map onto one

another. The shading is therefore wrong.

Figure 15. A recoloring of Figure 14 shows us that the

interior of the finite triangle projects to the union of an infinite

trapezoid and an infinite triangle. Where does the break

happen? The plane passing through O and parallel to the

image plane intersects the finite triangle in a line that gets sent

to infinity. (Compare with Figures 16 and 4.).

Figure 16. If we rotate Figure 15, we see something that

reminds us of Figure 4, as seen from above. In this figure,

though, the object triangle begins to dip below the image

plane.

12 THE MATHEMATICAL INTELLIGENCER

some of which show the projectivity from a more intuitivepoint of view.

A Matter of Viewpoint

As we have seen in the preceding section, it is a hard butworthwhile exercise to interpret Desargues’s diagrams asdrawings of two triangles in space, each in a separate plane,in which rays from a point O project vertices of one triangleonto the other. Readers who want to see how tricky this is

should cover up Figures 17 and 18 and try their hands at thediagrams with centers at C2 or E1! And often, as the pinholeexample shows, it’s helpful to have several meta-versions,some of which show the projectivity from a more intuitivepoint of view. Finding a ‘‘good’’ viewpoint, from which thediagram can easily be interpreted, makes all the difference.

For this paper, the authors made several (sometimesmany)pencil-and-paper sketchesof eachdiagram in trying to‘‘see’’ the figure. When we got a representation that pleased

(b)

(a)

Figure 17. The configuration for Coxeter’s diagram with the center at C2, two ways. In (a), we view the configuration as seen

from one side, with an eye indicating the viewing location for the image in (b), which matches Coxeter.

Figure 18. The configuration for Coxeter’s diagram with the center at E1, two ways: (left) as seen from one side; (right) rotated

toward us and as seen from above, matching Coxeter’s figure. The points P, Q, and R are labeled to assist with understanding the

rotation.

Figure 19. If we see the coordinate axes configured as they are on the left, we are in the main quadrant (where x, y, and z are all

positive). Where are we if we see the axes as in the middle or rightmost configurations? This is a question Pat Oakley asked her

students.

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us, we created a more formal version using drawing software(to be specific, Lineform, commercially available from Free-verse.) In this way, the process reminds us more of art than ofmathematics.

But of course, the mathematics drives these figures. Ourapproach to Desargues’s diagrams was motivated in part byan exercise that Pat Oakley of Goshen College cooked up forher Calculus 3 students in response to the Viewpoints math/art materials [12]. She had her students figure out where in R3

they were if they saw the axes in various configurations (seeFigure 19). It’s a great exercise!

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14 THE MATHEMATICAL INTELLIGENCER

Drawing on DesarguesDefinitionsDrawing on CoxeterA Matter of Viewpoint

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