Two-sided ruler constructions 1: Introduction

This blog post series is about geometric constructions using a tool that is two long parallel straight edges, basically exactly like a physical ruler, except without any marks to measure lengths with (or at least, you don’t use the marks). This tool is variously known as a parallel straightedge, a fixed-width straightedge, a double-edged straightedge, or a double straightedge. I like to call it a two-sided ruler, or just a ruler if you already know I’m talking about a ruler with two sides. (Though, I do have to say the phrase “double-edged” does have a certain appeal because it makes it sound like a sword.)

The reason I have to specify that the ruler has two sides is that in classical ancient Greek geometry, the tools were a compass (for drawing circles) and a one-sided ruler (for drawing lines). There are any amount of things you can read about classical ruler and compass constructions, and you can find out about a dizzying array of cool art and mathematics that can be made using them. However, you can construct all the same things with a two-sided ruler that you can with a ruler and compass (except the actual curve of a circle). Also it seems to me that rulers are much easier to come by and easier to manipulate with your hands than compasses, so I have come to love the idea of two-sided ruler constructions. Indeed, anyone who has been near me recently will likely tell you that I’ve been obsessed with them.

According to Florian Cajori [3], two-sided ruler constructions were comprehensively described by August Adler in 1890 [1], though that work is hard to come by and is in German anyway, so I haven’t read it. However, Sandra Kay Birrell did a great job of describing them in her 1983 Masters thesis [2] and I have read that. (The part on double-edged straightedge constructions begins on page 50.) She says this chapter was inspired by a paper by William Wernick [5], which actually has a few nice extra constructions that Birrell didn’t include. Chris Tisdell has done some videos on parallel straightedge constructions too [4], though I watched them long after beginning my own investigations.

In the blog posts that follow are my own idiosyncratic way of making sense of two-sided ruler constructions. It’s in a different order than those before me have done. Indeed, it’s actually been quite tough to put it all in an order that makes sense, and I have to say I fully believe now that Euclid’s most spectacular achievement when he wrote his Elements was deciding what order to put his propositions in.

Anyway, this presentation contains many constructions of my own design, some constructions I’ve modified from other people’s designs, and some of my own proofs of constructions thatr other people designed. You can assume anything where I haven’t explicitly said where it came from is mine.

[Note I am in the process of making photos and videos of all the constructions, so the blog posts will appear in stages as I make the media.]

Index of blog posts

Here are the blog posts in the series:

1. Introduction (you are here)
2. Fundamentals
3. Rhombuses
4. Copying and cutting
5. Perpendicular lines
6. Parallel lines
7. Circles without circles
8. Equilateral triangle and regular pentagon

Index of constructions and other results

This is a more comprehensive list of the constructions, definitions and other results mentioned in the blog posts, in case you (or me) are looking for something specific.

Introduction

• Definition of two-sided ruler
• References

Fundamentals

• Construction: Draw a line anywhere
• Construction: Draw a line through a specific point
• Construction: Draw a line through two specific points
• Construction: Draw two parallel lines anywhere (a ruler width apart)
• Construction: Draw a line parallel to an existing line (and one ruler width away)
• Construction: Draw a line a ruler width away from both of two points
• Definition of cross-aligning
• Definition of side-aligning
• The cross-align arcsine lemma
• Construction: Double a line segment (longer than ruler width)
• Construction: Double an angle

Rhombuses

• Construction: Draw a rhombus anywhere
• Construction: Draw a rhombus with a specific diagonal
• Construction: Draw a rhombus with a specific angle
• Construction: Draw a rhombus with a specific side
• List of properties of rhombuses
• Construction: Draw a right angle anywhere
• Construction: Draw a right angle anywhere (using a triangle)
• Construction: Bisect a line segment (longer than ruler width)
• Construction: Bisect an angle
• Construction: Bisect an angle (from the outside)

Copying and cutting

• Construction: Copy a line segment next to itself
• Construction: Copy a line segment along its line to an arbitrary point
• Construction: Double a line segment
• Construction: Bisect a line segment
• Construction: Copy a length from one arm of an angle to the other

Perpendicular lines

• Construction: Draw a line perpendicular to an existing line through a point on that line
• Construction: Draw a line perpendicular to an existing line through the middle point of three equally-spaced points on the line
• Construction: Draw the perpendicular bisector of a line segment by extending it both ways.
• Construction: Draw the perpendicular bisector of a line segment.
• Construction: Draw a perpendicular bisector to a line segment (using three parallel lines)
• Construction: Draw a perpendicular to an existing line through a point not on that line
• Construction: Draw a perpendicular to an existing line through a point not on that line (without drawing a rhombus)

Parallel lines

• Construction: Draw a line parallel to an existing line through a point not on that line (using a perpendicular line)
• Construction: Draw a line parallel to an existing line through a point not on that line (using three existing equally spaced points on the line)
• Construction: Draw a line parallel to an existing line through a point not on that line (using three existing equally spaced points on the line, and an intermediate parallel line)
• Construction: Draw a line parallel to an existing line through a point not on that line (using three parallel lines)
• Construction: Draw lines parallel and perpendicular to an existing line through a point not on that line
• Construction: Draw lines perpendicular and parallel to an existing line through a point not on that line

Circles without circles

• The trigonometry of lines meeting circles
• Division using similar triangles
• Construction: Mark the length r/d given the lengths r and d measured along the same arm of a right angle
• Construction: Mark the length r/d given the lengths r and d measured along different arms of a right angle
• Construction: Mark the length r/d given the lengths r and d measured along the same arm of a right angle (using a rectangle)
• Construction: Mark the length r/d given the lengths r and d measured along different arms of a right angle (using a rectangle)
• Construction: Find the points where a line meets a circle defined by its centre and one point on its circumfernce
• The trigonometry of circles meeting
• Construction: Draw three perpendicular lines to a line segment (longer than ruler width) at its two ends and at the midpoint
• Construction: Draw the perpendicular bisector of a line segment, given existing perpendicular lines through the endpoints.
• Construction: Mark the place where the line joining the intersection points of two circles meets the line joining the centres of the circles, given the radius of each circle measured at right angles to the line joining them (by reflecting a length)
• Construction: Mark the place where the line joining the intersection points of two circles meets the line joining the centres of the circles, given the radius of each circle measured at right angles to the line joining them (by reflecting an angle)
• Construction: Mark the place where the line joining the intersection points of two circles meets the line joining the centres of the circles, given the radius of each circle measured at right angles to the line joining them (by reflecting the radiuses)
• Construction: Find the points where two circles meet, each circle defined by its centre and one point on its circumference

Equilateral triangle and regular pentagon

• Construction: Draw a square
• Properties of an equilateral triangle
• Construction: Draw an equilateral triangle
• Construction: Draw an isometric grid
• Properties of the regular pentagon
• Construction: Draw a regular pentagon
• Construction: Draw a big five-pointed star
• Construction: Draw a small five-pointed star

References

[1] Adler, August. “Ueber die zur Ausfithrung geometrischer Constructionsaufgaben zweiten Grades not- wendigen Hilfsmittel.” Wiener Sitzungsberichte d. Akademie d. Wiss., Math.-Naturw. Classe, 99 (1890): 846-859.

[2] Birrell, Sandra Kay. “Euclidean constructions: alternate tools to the traditional compass and straightedge.” Master’s thesis, California State University, Northridge, 1983.
https://scholarworks.calstate.edu/downloads/0c483n19f

[3] Cajori, Florian. “A Forerunner of Mascheroni.” The American Mathematical Monthly 36, no. 7 (1929): 364–65. https://doi.org/10.2307/2298942

[4] Tisdell, Chris. “Geometry without circles! Meet the Parallel Straightedge” Youtube Playlist, 11 videos. https://www.youtube.com/playlist?list=PLGCj8f6sgswlijtwz1iiZCUe00DHH2x1p

[5] Wernick, William. “Geometric Construction: The Double Straightedge.” The Mathematics Teacher 64, no. 8 (1971): 697–704. http://www.jstor.org/stable/27958647