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[1] Dudley, Underwood, *A Budget of Trisections*, Springer Verlag, 1987.

This delightful book begins with a historical overview of angle trisections, using devices other than a straightedge and compass, such as Archimedes' angle trisection using a compass and a straightedge with two marks. He then describes the personalities of some would-be angle trisectors, then details dozens of angle trisection attempts.

[2] Eves, Howard, *A Survey of Geometry*, Allyn and Bacon, 1963.

This encyclopedic text is chock full of fascinating tidbits. In particular, Section 4.4 (pp. 198-204) discuss the Mohr-Mascheroni construction theorem which states that all Euclidean constructions could be carried out with a compass alone and no straightedge. Section 4.5 (pp.204-210) details the Poncelet-Steiner theorem, which shows that a straightedge along with one given circle and its center is sufficient to carry out any desired Euclidean construction. Section 4.6 (pp. 210-217) discusses other construction results, and mentions Lemoine's 1907

geometrography,a counting scheme for determining the complexity of a construction. Lemoine generally counts three operations where Hartshorne's construction counts one "step." We have followed Hartshorne's simpler counting method.

[3] Hartshorne, Robin, *Companion to Euclid*, American Mathematical Society, Berkeley Mathematics Lecture Notes, Volume 9, 1997

[4] Hartshorne, Robin, *Geometry: Euclid and Beyond*, Springer, Undergraduate Texts in Mathematics, 2000.

The 2000 title is an update of the 1997 version. Pages 20-22 in the newer version discuss the number of steps Euclid used for a proof versus the number needed for a mere construction. The homework following this section contains several constructions with an average or "par" estimate of how many steps an experienced geometer might use. Page 25, problem 2.14, asks for both of the trisection points of a segment and rates it "par 6". Of course, we are only asking for the left trisection point, so our equivalent is "par 5." Fascinatingly, in the 1997 version, the same problem 2.14, only now on page 23, says par=9. Either experienced geometers improved a lot in three years, or the original textbook had a typo.

[5] Heath, Sir Thomas, *The Thirteen Books of Euclid's Elements with introduction and commentary*, Dover Publications, 1956.

The bisection construction pictured is very close to that in Euclid's first proposition. Euclid actually does not prove the bisection construction until Proposition 10. (The relevant Propositions 1 and 10 are pages 241 and 267-268.) The bisection construction we have pictured is said to be due to Appolonius. Euclid clearly is more interested in the logical development of the proofs rather than in the shortest constructions.

[6] Hull, Thomas, http://kahuna.merrimack.edu/~thull/origamimath.html

Tom Hull has a fascinating set of origami geometric constructions, including how to trisect an angle. He has numerous books on origami and math, such as

Project Origami: Activities for Exploring Mathematics.

[7] Lang, Robert J., *Origami and Geometric Constructions*, pdf file available at http://www.langorigami.com/science/hha/origami_constructions.pdf

Robert Lang is a physicist who is an expert on mathematics and origami. In particular, he summarizes the best set of mathematical axioms for paper-folding, and discusses the most efficient ways to trisect a line segment. His web site http://www.langorigami.com/ has a beautiful collection of origami objects he has folded.

[8] Lemoine, Emile, *Geometrographie, ou Art des Constructions Geometriques*, Scientia, Phys.-Math. no. 18, Paris, February 1902.

Lemoine invented a method to measure the complexity of geometric constructions. His method has four parameters; one gives the required number of lines, a second the number of circles, while two others count the moves needed to place the ruler and the compass. Trisection is not explicitly mentioned in this monograph, although pages 34-36 give a more general construction, a corollary of which is essentially our third trisection method (see Reusch and Ringenberg below).

[9] Martin, George, *Geometric Constructions*, Springer, 1998.

This nice undergraduate geometry textbook explicitly develops the Mohr-Mascheroni and the Poncelet-Steiner constructions. We use his convention for "ruler points", pages 69-82, for the Poncelet-Steiner type of constructions.

[10] Nelson, Roger, *Proofs Without Words*, Mathematical Association of America, 1993.

Our first trisection is taken from page 13, attributed to Scott Coble.

[11] Reusch, J., *Planimetrische Konstruktionen in Geometrographischer Ausfuhrung, Druck and Verlag von B. G. Teubner*, Leipzig and Berlin, 1904.

Reusch expands on Lemoine's monograph with many diagrams and even more explicit analysis of the basic geometric constructions. Pages 17-20 explicitly deal with trisections; he gives three different constructions. The one he calls classical takes 4 lines and 6 circles; his second is the our third one a la Hartshorne that we are calling classical. Reusch's third construction uses four circles and a line (another "par" 5 construction). Here are scans of the relevant pages: V, VI, 17, 18, 19 ,20

[12] Ringenberg, Lawrence, *Informal Geometry*, John Wiley and Sons, 1967.

This standard text contains the typical method of trisecting a segment. Here is his version of the classical construction, page 139. In our third construction, we use two circles to construct his point C and draw the line

A C and then a third circle constructs C2. We do not need to draw C3 nor B C3 since the geometry is such that our line C2 D is already parallel to B C3. Thus, this slick version of the classical construction takes only three circles and two additional lines.

Our second and fourth trisection constructions do not seem to appear on the web or in standard modern texts, nor are our Mohr-Mascheroni and the Poncelet-Steiner trisection constructions explicitly shown anywhere. (The 12 line Poncelet-Steiner construction illustrated is due to Milos Tatarevic, who sent it in a couple emails dated June 7, 2003, and is used by permission.)

The third construction given, using three circles and two lines, is well known. Here is the typical diagram that is used to trisect the segment AB (due to Dr. Math at the Math Forum.) In our third trisection, the two initial circles construct the point C in Dr. Math's and Ringenberg's classical construction.

/

D o.

/ `.

/ `.

C o. `.

/ `. `.

/ `. `.

A o--------+---------+--------o B

`. `. /

`. `. /

`. `o E

`. /

`. /

`o F

/

Here are some trisection constructions that appeared in a Google search.

http://forum.swarthmore.edu/dr.math/problems/cbr.8.13.99.html

http://www.math.niu.edu/~rusin/known-math/98/trisect

http://www.cut-the-knot.com/arithmetic/rational.html#const

Using origami for geometric constructions can delight one for hours: Robert Lang has a nice discussion showing a trisection using four paper folds.

These emails from Milos Tatrevich discuss the general Poncelet-Steiner constructions: he has generalized the 1/3 construction for 1/n with beautiful combinatorics and conjectures how many lines are needed, for instance, to construct 1/7 probably takes 14 lines.

I received an email explaining how carpenters use a square and a ruler to trisect line segments:

Subject:

Date:

Sun, 23 Feb 2003 20:36:11 -0500

From: "T. Wilson"

Enjoyed your methods of trisecting a line segment. Now if the criteria include the use of only a compass and an unmarked straightedge, then the method I was taught as an apprentice carpenter would qualify, wouldn't it?

Take line AB. Extend two parallel lines perpendicular to A and B. Take the straightedge and mark off four equidistant points along one edge. Place the straightedge at an angle between the parallel lines so that the first and fourth points coincide with the parallel lines drawn from A and B. With the compass determine the perpendicular distance of each of the three middle points from either parallel line and transfer that distance to the line AB. Voila.

Simple strokes for simple minds (mine, of course, not yours!).

Best wishes, T. Wilson, Richmond, VA

Note that carpenters use a marked ruler, which opens a fascinating world of constructions going beyond Euclidean, in particular, one can use a marked ruler to trisect angles.

I received another email with a beautifully symmetric construction using two circles and four lines:

Subject: Trisecting a line segment

Date: Sun, 3 Aug 2003 12:19:14 -0500

From: "Fred Barnes"

To:

I just stumbled across your wonderful web page on trisecting a line segment. I've included a method, not on your page, using four circles and two lines. See attachment.

Fred Barnes

Robert Styer (Villanova Univ.), "Trisecting a Line Segment (With World Record Efficiency!)," *Loci* (February 2010), DOI:10.4169/loci003342