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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El Zapatilla Walker Esta diseñado para mantenerte en movimiento con la máxima comodidad y soporte.
Completo grano resistente al agua cuero con contraste superposiciones para un atractivo deportivo.
Acolchado collar y lengua con tradicional cordón proporciona un cómodo, ajuste personalizado.
Transpirable, absorbe la humedad forro de malla con La base de espuma PU ayuda reducir la fricción contra la piel.
Contorneado plantilla ortopédica con tacón profundo taza estabiliza, soportes y realineaciones los pies de regreso a su posición natural Ligero, flexible densidad media La entresuela de EVA absorbe choque para reducir estrés en los pies, tobillos y rodillas. Firme, termoplástico contador de talón proporciona mayor estabilidad y apoyo para todo el día. suela de goma con surcos flexibles proporciona tracción y flexibilidad. Recibió el sello APMA de aceptación. Haga clic aquí para ver Sello. Peso: 15 oz
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,

Sandalias de cuero de Viv Viv de TOMS Cuero de Miel / Emossed Melissa Shoes Womens Wonderful + Jason Wu Desnudo / Beige , CLARKS Womens Un Plaza Sling Peach Nubuck Sandalia Rosa mujer TOMS Ante negro Taipán de mujer de David Tate Verona Cordero de plata , Sandalias con plataforma SAS de cuero con punta abierta para mujer Negro , Sandalia de cuña Daytona SoftWalk para mujer Azul marino W Tan Comfortiva Mujer, Faye Sandalia de tacón medio Rojo rubí , Alegria Womens Morgyn Sandal Cosas lindas Chaco Mujer Zcloud 2 Sport Sandalia Verdure aguacate , Sandalia de Richetta de Vince Camuto para mujer Negro OLUKAI Pua Sandal - Mujer Deep Red / Bean Sandalia Uneek-W de mujer KEEN Midnight Navy / Cendre Blue , Sandalia mujer rosa superfeet Negro CLARKS Mujer Gracelin Art Mary Jane Flat Cuero Tan oscuro , Bomba de plataforma Camper Women s Damas 21923 Marrón medio , Sandalias EVA Gizeh de mujer Birkenstock Eva Cobre Metálico , Sandalia Dionay Calvin Klein para mujer Rojo carmesí BareTraps Sandalia para mujer Octavia Armada Caja amarilla para mujer Sullivan gris Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Champagne / Metálico Sandalia de puntera de cuero ajustada FitFlop para mujer Negro Teva ALP Premier - Mujeres Armada Sam Edelman sandalia Gio Slide para mujer Ante negro , Sandalia de mujer KEEN Rose Boysenberry / Red Violet Sandalia CLARKS Tri Clover para mujer Black Combi , Diapositiva trenzada de Aetrex Janey Bronce , Sandalia de cuero Terra Fi Lite de Teva para mujer Negro Sandalia plana Nanci de Naturalizer para mujer Mezclilla ,
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 viaje Protic W3910 Vista Walker para mujeres Blanco suave , Zapato Casual de Cuero Cortez Clásico NIKE para Mujer Pure Platinum / Pure Platinum Nike Men s Air Presto Essential Negro / Negro-negro NIKE Mujeres Benassi Duo Ultra Slide Womens 819717-002 Blanco negro Inov-8 Women s X-Claw 275-W Trail Runner Negro / verde azulado / Berry , Sandalia ECCO Women s Offroad Lite Roca lunar , Pico Azul , Zapatillas de running Reebok para mujer Collegiate Royal / White , Sandalias Harper Huarache de SoftWalk para mujer Beige / Tan , Drew Shoes - Cremallera de cuero con bota de profundidad extra diabética terapéutica para mujer Athens Marron oscuro , Zapatillas de baño Skechers Performance para mujer Go Step-Seashore Natural CLARKS Sandalia de cuña Arla Jacory para mujer Sólido negro , Vivien Flat de Bella Vita para mujeres Almond Kid Suede Leather , Zapato de entrenamiento WX824 para mujer New Balance Negro Vans Auténtico Deep Orchid / True White , Vans Herren Authentic Core Classic Sneakers True White Clásicos de base Slip-on (Tm) para hombres de Vans Armada , Zapatillas NIKE Roshe One Nylon para mujer Coral Stardust / Blanco Sandalia tobogán X-Band con plataforma Mikenzie de Gentle Souls para mujer Seta Sandalia Buxom Slide de Charles David para mujer Blanco Sandalia Dolce Vita Halle Slide para mujer Tela natural / negra , Sandalia de cuero Nirveli de Nine West para mujer Cuero natural oscuro , Sandalias de Jerusalén Sandalias de mujer Edna rojo Rieker Mirjam 11 bomba para mujer Beige Zapatillas Snowbird Ii Slipper de Old Friend para mujer castaña , Zapato borroso de mujer de Steve Madden Bright Multi , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Wolf Gray Black Volt 013 , TOMS Women Classic Slip on Black.on.black , Easy Devitt Oxford Flat para mujer Azul , Zapatillas de deporte Janet Calvin Klein para mujer Cuero 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 Tecnología Vionic Men s Orthaheel White/ Navy B01M8PLGT1 Walker Navy Vionic Negro 0ba77c9 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 744897
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.