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.

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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

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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

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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.

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

Bomba de vestido de pasión Easy Street para mujer Nueva Marina Trotters para mujer Dea Slingback Pump Armada Bomba de vestido Daisie de mujer Steve Madden Ante azul J. Rene233; Gris pardo Zapatilla de cuña para mujer Jacques Levine # 4640 Blanco metálico , Mujeres de Modemoven Charol Bombas, Point Toe Tacones, Slingback Bombas, zapatos de noche, lindo Stilettos 10CM Amarillo , Bomba de vestido asimétrico para mujer Aravon Black Leathe Calzado de vestir Genoveva para mujer Calvin Klein Blanco suave , Pantuflas Queen de mujer verde de Patricia Natural , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Corduroy de arena , Zapatos de tacón alto con tachuelas y tachuelas Kaitlyn Pan en punta Patente negra / correas negras / tachuelas de pistola de metal TOMS Clásicos de mujer Ceniza , Ciabatta s genuina Shearling rayado deslizamiento deslizamiento zapato deslizador (bronceado) , Salón muy fino Salsa de tango latino Zapatos de baile para Mujeres 1620 1.3 pulgadas Talón + Plegable Paquete de cepillo Satén marrón Zapatilla Peluchey Flexus de Spring Step para mujer Negro , CLARKS Bomba de cuña Flores tulipán para mujer Serpiente de piedra , Zapatillas Laurin Moccasin de Tempur-Pedic para mujer Negro , Zapatilla de lana para interiores / exteriores Therafit Heather para mujer Broncearse , Zapatilla de mujer TOMS Oro blanco Glerups modelo unisex GR Slipper Gris Sam Edelman Women s Addison Sandalia de tacón Desnudo clásico Calzado de vestir Genoveva para mujer Calvin Klein Platino blanco , Zapatilla deslizante Loki II para mujer BEARPAW Pom Berry , Zapatilla de UGG Women s Lane castaña Zapatilla Glamour de mujer verde Daniel Bone Kidskin Zapatillas Vionic Pleasant para mujer con soporte Arch - Orthaheel Technology Vino LifeStride Women s Lively 2 Pump Sonrojo Sandalia de vestir Carrson de mujer Steve Madden Lavanda de gamuza , Trotters Lucy Ballet para mujer Desnudo
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 Sandalia de Hollyleaf de las mujeres de Merrell Fucsia , Chaco Mujer Zcloud X Athletic Sandalia Acción azul , Chaco Women s Zx2 Classic Athletic Sandal Tunnel Tango Sandalia de mujer Drew Alana Blanco negro , Converse Chuck Taylor All Star Seasonal Color Hola Beach Glass Rieker Womens 41346 Doris 46 Blanco apagado / Metálico / Hielo Easy Spirit Solana Oxford para mujeres Gris Zapato para caminar WW877 para mujer New Balance Plata , Sandalia Dominica de Ryka Gris / morado Zapato de agua nativo de Lennox para hombres Regatta Blue / Shell White Converse Chuck Taylor All Star Core Ox Kombu Green Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre Verdant Green , Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre True White , PUMA x Fenty enredadera entallada Puma Black , Charles by Charles David Sandalia de cuña Brooke para mujer Microsuede gris claro Calvin Klein Dalana Slide Sandalia para mujer Negro / Peltre / Blanco Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla Sandalias Ava Sandalias Sandalias de Jerusalén Blanco Zapatillas Dr. Mary Comfort Flauta Lycra Elasticables Diamante Mary Jane Licra Ariat Womens Devon Pro VX Paddock Negro Calzas de la dinastía Azul marino / blanco , Drew Women s Summer Negro , Junk Gypsy Western Boots Mujeres Spirit Animal Fringe Tobillo JG0040C Crema , CLARKS Mujer Adya Bella Tobillo Botín Ante negro , Nurse Mates Womens Fleet Negro , Bota Deia para mujer de TOMS Cuero negro , Bota para mujer Hana de Zerogrand de Cole Haan Arena Under Armour Women s Lax Finisher MC Blanco / anticongelante Trotters Liz Loafer para mujer Armada TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Negro

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.

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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 Dualyz Fit 4648 Unisex claro 100% lana Zapato Fit con suela extraíble Gris claro/ negro b52ed82 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 1181942
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.