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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Sintético y malla
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,

Zapatos de caminata SAS Low Me de cuero para mujer Negro , Calzado para caminar de mujer Mephisto Cross Chanti Bucksoft / Snake / Wine Patent Zapato de malla Nohea de OLUKAI - Hombre marrón Zapato de malla Nohea de OLUKAI - Hombre Rock / Canoa , Zapatilla Ascot de hombre UGG Musgo Verde , Tierra Everwood Cuero de becerro negro , Zapatillas de ciclismo X-Project 2.0 Pearl iZUMi para mujer Negro Negro , Easy Spirit Women s Traveltime Mule Blanco , Zapatillas de deporte vanguardista Speeds Varsity Vanguard para mujer Brooks Riviera / Microchip Gray , Zapatos de caminata SAS Low Me de cuero para mujer Negro , ALLROUNDER by MEPHISTO Zapatillas Mujer Fina Tex Sport Goma negra / nubuck rojo medio , Zapatillas de bochas ancho Dewalter T.H.E 9 Dexter para mujer Blanco / Lila Zapato de mujer Jessica Simpson Fitt Walking Estrella fugaz , Zapato de malla Nohea de OLUKAI - Hombre Carbón de leña / Dive Camo Tierra Everwood Khaki Suede / resistente al agua Sandalias Sunstone para mujer Merrell Vertiver Easy Spirit Women s Kandance2 Flat Tela de vino , Zapato de mujer Jessica Simpson Fitt Walking Blanco negro Easy Spirit Women s Traveltime Mule Blanco / Verde oscuro / Cuero Multi , Easy Spirit Women s Traveltime Mule Patente negra / plateada Zapato de malla Nohea de OLUKAI - Hombre Blanco VANELi Womens Jaicee Ante negro / elástico negro Under Armour Women s Squad Negro / Blanco / Blanco ALLROUNDER by MEPHISTO Zapatillas Mujer Fina Tex Sport Nogucho negro / caucho negro , Zapatilla Ascot de hombre UGG Sáhara , Zapatillas de deporte vanguardista Speeds Varsity Vanguard para mujer Brooks Riviera / Microchip Gray , Zapato de malla Nohea de OLUKAI - Hombre Trench Blue / Dive Camo Baabuk Wool Sneaker - Mujer Azul marino / blanco ALLROUNDER by MEPHISTO Zapatillas Mujer Fina Tex Sport Negro / Curry ,
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 Wolky Comfort Jewel 80530 Charol Rojo Coral , Under Armour Women s RailFit 1 Clay Green (104) / Antracita , Salomon X-Tour Light para mujer Wasabi / White / Mystic Purple Sanita Sanena Women s Deena Desnudo , Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Turquesa , Zapatilla deportiva Kailee de mujer Ryka Frost Grey / Cotton Candy / Summer Grey , I-RUNNER Maria Puntera terapéutica para mujer con profundidad extra Calzado de cuero / malla con cordones Gris / Salmón Sandalia de cuña Reeta para mujer Lauren Ralph Lauren Geranio rosado Sandalia de cuña Logan Walking Cradles para mujer Desnudo Flip-flop Oliver de Sam Edelman para mujer Patente de Turquiose Sandalias de tacón bajo Pikolinos para mujer, Mykonos Negro , Dr. Comfort Lu Lu Profundidad terapéutica para diabéticos terapéuticos Mujer Velcro de cuero para zapatos castaña , Zapatillas para mujer Del Rey TOMS Tela escocesa negra / blanca Sandalias Sandalias Bashan moldeadas Sandalias Sandalias Jerusalem Blanco , Sandalia Mason para mujer Naturalizer Slide Mármol , Sam Edelman Sandalia mujer Shaye Slide Negro , Zapatillas de voleibol adidas Crazyflight X 2 de mujer Choque amarillo / ceniza plateado / blanco 50 sombras de blanco Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 3540- Satén blanco , ALDO Mujeres Elinna. Patente negra 3N2 Mofo Turf Trainer Blanco Bomba de vestido Pointe para mujer Easy Street Patente roja , Vionic Womens Elevated Becca Wedge con cordones Bronceado suave Burton Limelight BOA Snowboard Boots para mujer , Calvin Klein Women s Gayle Pump Patente negra , Caminante de Flyroam de las mujeres de Timberland Jet Black Dt Suede , SOREL Bota de cuero baja grande para mujer Quarry / Shark Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Diabético ortopédico Pies planos de juanetes Zapatos de mujer - Wichita Negro , Botas de campo acolchadas TuffRider Ladies Alpine en piel sintética - , B.O.C., Peggy CHOCOLATE DE LA MUJER CON ACEITE 8 M - Zapatillas Slip-On Roamer SAS para Mujer Sabio ,

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 Zapatillas deportivas Zapatillas Under Armour Xc Brigade Brigade Spike deportivas Blanco negro 1e6fb68 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 3849593
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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.