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,

Carro de mujer Brooks Negro , Brooks Womens neuro 3 Laguna / Negro / Púrpura Brooks Womens mazama 2 Rosa / ciruela / azul marino Zapatillas deportivas 574S Sport New Balance para hombre Gunmetal / acero , Zapatillas de running Strobe V1 para mujer New Balance Blanco Zapatillas deportivas 574S Sport New Balance para hombre Azul marino / Vintage Indigo , Converse Chuck Taylor All Star Seasonal Color Hola rojo , Converse Chuck Taylor All Star Seasonal Color Hola Pistola Metal / Blanco , Zapatillas deportivas 574S Sport New Balance para hombre Gunmetal / Navy Calzado casual tejido NIKE NIKE para hombre Negro , Zapatillas de deporte Onitsuka Tiger Mexico 66 Blanco / azul oscuro , Under Armour Women s HOVR Phantom NC Carbón de leña (104) / Gris opaco Zapatillas deportivas 574S Sport New Balance para hombre Nimbus Cloud / Navy Zapatillas de running NIKE Juvenate para mujer Apenas Verde / Esmalte Verde / Vela , Zapatillas de deporte de moda Kanye West Desert Rat 500 Zapatillas de correr 3M para hombres Mujeres DB2908 DB2966 Super Moon Yellow , Zapatillas de running Ultra Kane 3.0 de Zoot para mujer Gris / Verde Flash / Atomic Blue Zapatillas de running NIKE Roshe Two para mujer rojo , Zapato Casual de Cuero Cortez Clásico NIKE para Mujer Mushroom / Summit White / Gum Medium Brown / Mushroom / Multicolor Zapatillas NIKE Femenil Flex 2016 Rn Running Pure Platinum / Black White , Under Armour Women s Speedform Slingride 1.1 Negro (001) / Blanco , Zapatillas de running Spira para mujer Stinger XLT Uva / Lima / Blanco , Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Lingrn, Lingrn, Tacgrn , Zapatillas de running Saucony para mujer Peregrine 8 Negro / Menta Reebok Classic Sneaker Blanco / gris claro Carro de mujer Brooks Cerámica / Aguilera / Azul Iris , Converse Chuck Taylor All Star Seasonal Color Hola Rosado , NIKE Kids gratis Rn (grande) Universidad Rojo / Negro-blanco Converse Chuck Taylor All Star Seasonal Color Hola Pergamino / negro / blanco La Sportiva Lycan Running Shoe - Mujer Stone Blue / Jade Green ,
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 Saucony Women s Jazz Low Pro Azul blanco , Zapatillas de running EM Tri N1 v2 para mujer Pearl iZUMi Blue Atoll / Electric Pink , Sandalia KEEN Women s Venice H2 Cascade / Misty Jade Sandalias KEEN Venice II-W para mujer Aceite de visón / Tierra oscura Zapatillas para correr A7200WX085 de Apex para mujeres Rosado Zapatillas de running Reebok Hexaffect 5.0 MTM para mujer Gable Gray / Vitamina C / Blanco / Negro Zapatillas de running ASICS Gel-Cumulus 19 para mujer Azul Púrpura / Negro / Flash Coral Zapatillas sin cordones Flex Merit para mujer Merrell Profundidad azul , Walking Cradles Womens Henna Nubuck negro estampado , Zapato de running ASICS GT-1000 6 para mujer Negro-negro-plata Mocasín de mujer RIALTO Vette Azul marino / E-print , Skechers Performance Women s Go Run 400 Obstruct Negro Sandalia Pamela Mujer Mephisto Negro , Sandalia de Gladiador de Asha de mujer Ahnu Oxblood CLARKS Janey Mae Oxford para mujeres Cuero negro , Vans Unisex Old Skool Zapatillas clásicas de skate Negro , Lakai Fremont Skate Shoe Grey Suede Zapatillas de skate Vans Unisex Authentic Platform Púrpura (Petunia True White Qa1) 7270 Sandalia de tobillo con plataforma FitFlop Women s Banda de cuero con tachuelas Rosado oscuro , Sandalia con tiras metálicas Violet Shimmer de Taryn Rose para mujer Oro , DAWGS Womens Arch Support Sandalias Loudmouth Z Shagadelic Black Calzado Aria Pointe 121 para mujer Capezio Pétalo rosa , Noble Outfitters Womens Muds Stay Cool 6 Deep Turquoise Marc Jacobs Sienna Espadrille Ballet Mujer plana Gold Multi , Trotters Lucy Ballet para mujer Black Suede / Patent Zapatos muy finos EK8881 de la danza de salón de señoras de las señoras con 1 \ , Mujeres muy finas Ballroom Latin Tango Salsa Dance Zapatos 2707 2.5 \ Talón y plegable Paquete de cepillo para zapatos Satén blanco NAOT Kedma Oxford para mujeres Mine Brown Leather , Botas con cordones originales Sissinghurst de Hunters Boots para mujer Azul marino / goma , CLARKS Women s Marigold Slip-On Loafer 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.


Linkages

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 Zapato 19971 Blanco NIKE Zapato Roshe One femenino Blanco Blanco df7f42a it out, and I got a chance to try out this new graphics software.


Resources


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.

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

Geometry Construction Reference

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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.