The Tomahawk - an Angle Trisection Tool

by mkaercher

6.6K4124

Intro: The Tomahawk - an Angle Trisection Tool

While it is possible to bisect an arbitrary angle using only a compass and a straightedge, it is impossible to trisect an arbitrary angle with only a compass and straightedge. For more about this discussion, check out this Wikipedia post.

There are, however, a few tools that can be used to trisect an angle. One of those tools is called a Tomahawk. I stumbled upon a different Wikipedia post https://en.wikipedia.org/wiki/Tomahawk_(geometry) about this tool. I consider myself a proficient teacher and student of geometric constructions. So when I came across this interesting tool, I set out to make one and put together this Instructable.

STEP 1: How to Bisect an Angle - a Refresher


There are many ways to bisect an angle, using a compass and straightedge. Here are two of those ways. This will give you a little context before we make our Tomahawk.

STEP 2: How to Make a Tomahawk


To make the Tomahawk, you will need some paper, card stock, or cardboard. I will demonstrate using a 3" x 5" index card. The dimensions work nicely for the design of the tool.

I based the design on the plans found in this article.

STEP 3: How to Use the Tomahawk


Now that you've made your Tomahawk, draw some angles and try using the tool to trisect the angles. Verify the trisections using a protractor.

The tool should work for any angle less than 180°. You can also make the Tomahawk smaller or larger, depending on your angle needs.

Also, if you do a little searching, you can also find publicly-shared Tomahawk files for the 3-D printer.

STEP 4: Final Thoughts

I love the simplicity of this tool and can't wait to have my students make it and try it out. I've also thought about renaming it an "Angle Axe".

I might even have students make one using compass and straightedge constructions. So even though the angle isn't trisected using a compass and straightedge, at least the tool was built using them.

I hope you've found this Instructable helpful and interesting. If you have any feedback, questions, or suggestions, please feel free to share them below. Thanks!

STEP 5: 3D Printed Version

Since publishing the original post, I thought about making a 3D printed version of the tool. I started designing the tool in Tinkercad, using a composition of shapes. But I couldn't get it just right. So I designed the shape of the tool using this Desmos Calculator. This allowed me to make the proper vertex where the handle meets the curve of the blade.

I then exported the PNG from Desmos and converted it to an SVG, allowing it to be imported into Tinkercad. Here's the link to the Tinkercad model: https://www.tinkercad.com/things/8A1TV5hWxGm

Note: As of this update, I haven't yet printed the model. When I do, I will be sure to update this project.

8 Comments

Mahesh R Bubna 1 year ago

The nearest solution to 2200 years old Geometric problem using only a compass and an unmarked straight edge....

GoFish 2 years ago

Looks like it will be a very handy addition to the toolbox. Thanks.

Mahesh R Bubna 1 year ago

I have trisected some angles to an unprecedent/ miraculous approximations as can be seen in the attached images.... (using only a compass and an unmarked straight edge.... with 9. 10 finite and easy steps

charlessenf-gm 2 years ago

Thanks, enjoyed watching that.
It appeared to me that the ratio was 3 x 4 rather than 3 x 5
You cut the top inch off the card, leaving 3 x 4 for the tommyhawk.
So Start with material where width = .75 Height and, then the radius Radius = Width/3 right?
Height = 4, width = 3, radius = 1

iamchrismoran 2 years ago

Oof... I didn't realize you couldn't trisect an angle, as I severely mistakenly let myself believe that trisecting a chord would let me trisect the central angle.
So, thanks for this instructable if for anything, to correct me.
Cool tool!

mkaercher 2 years ago

Thanks! It took a while for me to realize that same mistake.

Enjoy!