Whistler Alley Mathematics

These are some mathematics investigations I have pursued over the years. They may be of some interest to teachers, students, or hobbyists. I try to convey a conceptual understanding, usually without rigorous proof. Some of the lessons are accompanied by questions and suggestions for extensions.


%Article_Desc% Some of the lessons are accompanied by questions and suggestions for extensions.

In my pursuit of a teaching career, I was told to justify the study of each concept by establishing its relevance to the students’ lives. Sorry, but I still have trouble buying into that one. Fortunately, poetry and music are rarely put to that same test. As I glance at the list below, I must concede that it would be difficult to convert any of the lessons into food, shelter, or money. These are things that interested me, and now I understand them better. If there is a reward, it is the fact that every time I do this I get better at figuring things out.

I occasionally get email from people seeking permission to use my work. Feel free to cite, quote, edit, reproduce, or distribute anything on this site. Do not worry about putting my name on it, but please do not put your own name on it.

Many of these presentations use the Geometer’s Sketchpad, but only one of them actually requires it. If you do not have the software, let me suggest that you follow this link to Key Curriculum Press and look into it.

Paul Kunkel


Recent Changes

March 16, 2016

Updated the animations on Hanging With Galileo.

October 23, 2015

Updated the animations on Reuleaux Triangle.

October 9, 2015

Updated the animations on the Vernier Scale page of the Geometry of Surveying.

October 8, 2015


Updated the animations on Linkages.

October 4, 2015

Updated the animations on On the Square.

September 25, 2015

Updated the animations on Tangent Circles.

September 23, 2015

Updated the lead page of Conic Sections and Construction.
The page it replaced had java animations that were not being treated well by modern browsers. New images were generated with the developmental Web Sketchpad, which is producing very satisfying results. This appears to be the future, so expect more of my pages to be similarly updated soon.

February 26, 2015

Added Book VII to the Conics of Apollonius.
This is the final surviving book of the treatise.

October 22, 2014

Added Book VI to the Conics of Apollonius.

June 10, 2014

Added Book V to the Conics of Apollonius.

 Includes GSP-3 files

 Includes GSP-4 files

 Includes GSP-5 files Contents:

The Brachistochrone

We are given two fixed points in a vertical plane. A particle starts from rest at one of the points and travels to the other under its own weight. Find the path that the particle must follow in order to reach its destination in the briefest time.

This is a famous problem in the calculus of variations. Nothing new is presented here, but the explanation is more thorough than most.

Buffon’s Needle

An old probability exercise is aided by an interactive Sketchpad file. By manipulating the sketch, an intuitive understanding may be gained, even if the student has had no introduction to calculus.


Chinese Handcuffs This has no lesson, only questions. It is just an interesting animated file, which invites student investigation. It has applications for geometry,

trigonometry, and probability. Fair warning: The discussion revolves about a figure that appears only when the sketch is animated. Without Geometer’s Sketchpad, this one will make little sense at all.

Conic Sections and Construction

My emphasis is on geometric properties of the conic sections, particulary understanding that has been handed down to us in ancient texts. There also are some challenging constructions.

The Geometry of Surveying

This used to be my job, and I enjoyed studying all of the geometrical concepts that were part of the science of land surveying. In this article I discuss the instruments of the profession, their operating principles, their error, and their calibration.

Hanging with Galileo

An investigation into the catenary, the curve formed by a chain suspended at both ends. It also extends into the curve of the main cables on a suspension bridge, which, interestingly, is not a catenary.

Inversion Geometry Zapatillas de running NIKE Women s Air Max 1 Ultra Essentials Negro / Negro / Mtllc Slvr / Blanco Zapatillas de running ASICS Gel-Cumulus 19 para hombre Primer rojo / negro / plata Sandalia Romika Women s Hollywood 04 Sport Castagne OluKai Kulapa Kai Papaya / Negro , Sandalias deportivas Qianling Collection para mujer 80% PU / 20% sandalias deportivas de malla Azul Sandalia Drew Angela para mujer Beige-multi , Sandalia Archie Fisherman del Dr. Scholl Nubuck azul marino , Zapatillas de running adidas Women s Rockadia W Trail Gris / negro / rosa , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Azul / iridiscente , EN Zapatilla Cloudster para mujer Magenta / Berry , YDN Mujeres Open Toe High Heels Mules Slip on Bombas Diapositiva Transparente Vestido Sandalias Zapatos Gris CLARKS Danelly Millie Sneaker para mujer Blush Pink Leather , Botín para mujer Pleaser Asp1018 / Bpu / M Pat negro, negro Zapatos atléticos para diabéticos Meghan Turquoise de Dr. Comfort para mujer Azul , Vans Old Skool Unisex Adults Low-Top Trainers Mantequilla de manzana / True White , Supra Scissor Skate Shoe Negro-blanco-negro , Sandalia para mujer Cava Donald J Pliner Black Geo Jungle Sandalia Chloe Slide de KENDALL + KYLIE menta Sandalia Tirra para mujeres Teva Blanco negro , Charles by Charles David Sandalia para mujer Nicki Slide Desnudo Sandalia de la diapositiva de Stella de las mujeres de Rachel Zoe Plata Zapatilla Slip-On de Ahnu para mujer Portero , Vionic Women s, Josie Low Heel Pumps Negro Zapatillas de deporte de verano Breezy Dualyz Fit Breezy con suela extraíble Gris claro / gris claro , Azul de Betsey Johnson Zapatillas SB-Haven para mujer Sonrojo , Zapatillas de ciclismo Smartodoors con suela de carbono o suela de tpu de nylon para carretera y MTB Azul para MTB , XYD mujeres Slingback medias gatito talón bombas punta estrecha hebillas zapatos de vestir con elegantes arcos Pasto verde Vionic Women s Tide Ii Bronce Metálico Zapatillas de deporte de verano Breezy Dualyz Fit Breezy con suela extraíble Negro Negro , Zapatillas de deporte de mujer Ettana de Tommy Bahama Blanco (lino)

If you have never studied inversion geometry, you can start here. The elementary concepts are explained. This page is a quick reference for the other lessons that employ inversion geometry.


Draw a line segment. Did you use a straightedge? How do you know it was straight?

Java simulations are used to investigate a nineteenth century mechanics problem.

On the Square

The challenge: Given four arbitrary, unique points, A, B, C, and D, construct one line through each point such that the four lines form a square.

Orthocenter What is so significant about the orthocenter of a triangle? Why does it even have a name? This investigation was prepared for a geometry teachers’ workshop. It includes a lesson handout in Word format. This lesson was written for use in a computer lab with the Geometer’s Sketchpad, but with a little editing it should work just as well with Cabri or Cinderella.
The Planimeter

A planimeter is a drafting instrument used to measure area. Its mechanism is quite simple, but understanding why it works is more elusive.

Platonic Solids

Find the inradius, circumradius, dihedral angle, surface area, and volume of all of the Platonic solids. There is also some information about some of their other interesting properties.

Reuleaux Triangle

It is not a circle, but it has constant width. Study the properties of this figure with interactive animated sketches.

Rungs and Vineyards

How is that for a descriptive title? Here are investigations of two interesting visual effects seen from a moving automobile. It includes two interactive perspective drawings, and a discussion of the cylindrical and planar projections used to create them.

The Sliding Triangle

An interesting accidental discovery – Investigate ellipses by tracing triangle transformations.

Table Patterns

Explore geometric patterns formed in a table of modular multiplication. This one requires some elementary knowledge of congruence classes and modular arithmetic.


Tangent Circles

Three circles are given. How many circles can be constructed tangent to all three? How are they constructed? This lesson includes ten animations covering all of the special cases. Some of them are very complex. A little knowledge of circle inversions would help.

Volume of a Torus Do you already know the volume of a torus? Then humor me. I just figured %Article_Title% it out, and I got a chance to try out this new graphics software.


The Geometer’s Sketchpad Workshop

It is plain to see that I am very fond of Sketchpad. This page was created after the release of version 4. The idea is to have a place to exchange ideas with other users, especially those who are using the advanced features. Here are some custom tools and some information on Java Sketchpad.

Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 2369207
and compass constructions are described and illustrated. The original version, in Word format, can be downloaded and distributed.

Back to Whistler Alley

Last update: March 16, 2016 ... Paul Kunkel whistling@whistleralley.com
For email to reach me, the word geometry must appear in the body of the message.

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