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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tela y sintético
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,

Zapatillas Sequoia para mujer Mark Nason Angeles Lila NIKE Men s Roshe Run Olivo medio / negro / secoya / gris pálido Vibram Furoshiki Phulkari Sneaker para hombre y mujer Flor azul , Brooks Womens Revel Joyous Mint / Pink / White , Zapatillas Spira Aquarius Running para mujer Azul Zapatillas de running Fila Fresh s Memory Fresh Start 2 Slip Resistant Blanco, blanco, blanco , Zapatillas de senderismo adidas AX2 CP para mujer Vista Gris / Negro / Super Blush S17 , Zapatillas de running Skechers GOrun Forza 2 para mujer Azul marino / Aqua NIKE Women s WMNS Zoom Fly, negro / blanco Blanco negro , NIKE Kids gratis Rn (grande) Black Hot Punch 003 Zapatillas de running Adidas Performance Ultra s Boost M para hombre Azul / Misterio Azul / Negro , Zapatillas de running NIKE Roshe Two para mujer Wolf Gray / Wolf Gray , Under Armour Women s HOVR Phantom NC Marfil (105) / Acero , Converse Chuck Taylor All Star Seasonal Color Hola Limón amargo Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Clay Green / Barely Gray 300 , NIKE Boys Star Runner (GS) Gris Converse Chuck Taylor All Star Seasonal Color Hola Negro / rojo / blanco , Zapatillas para correr Mizuno Women s Wave Daichi 2 True Blue / Micro Chip / Blue Topaz Zapatillas de running NIKE Juvenate para mujer Orquídea / Orquídea / Midnight Navy White NIKE Women s Tanjun Negro / Metálico Rojo Bronce , Zapatillas Nike Womens Air Max 2015 Negro / Blanco / Cumbre Blanco / Refleja Plata , Converse Chuck Taylor All Star Seasonal Color Hola Blanco / Vapor Rosa / Blanco , Converse Chuck Taylor All Star Seasonal Color Hola Carbón Zapatillas de baloncesto Jordan Kids Jumpman Team II Negro / infrarrojo 23-cool gris-infrarrojo 23 Brooks Womens PureGrit 6 Negro / Ebony / Diva Pink , Converse Chuck Taylor All Star Seasonal Color Hola Ratón , NIKE W Zoom All Out Low 2 Womens Aj0036-200 Dusty Peach / Mtlc Red Bronze , Zapatillas de running Spira Phoenix para mujer con resortes Royal / Black / White , Zapatillas de running New Balance Fresh Foam Lazr V1 Melocotón espolvoreado / Melocotón espolvoreado
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 BZees Sandalia con cuña de beso para mujer Negro , Soulsfeng para hombres Moda femenina Zapatos deportivos de encaje up Amortiguación transpirable Diseño de flores de tela Zapatillas de deporte para parejas Blanco Zapatillas adidas outdoor Terrex AX2R para mujer Ch Gris / Dgh Solid Gray / Clear Aqua , Zapatillas para correr Salomon Women s Speedcross 4 GTX Black / Black / Metallic Bubble Blue 10 - , Zapatilla de deporte Rylee Fashion para mujer Chiara Caruso , Zapatillas de running Nike Zoom para mujer Hasta / Mtlc Red Bronze / Green Glow Atletismo femenino de Apex Negro , Zapatillas para caminar ProFit TravelFit Slide para mujer Oro / negro , Sandalia de cuña Tobacco Sugar Too de Jambu para mujer rojo , Sam Edelman Womens Daryn Golden Caramel Diva Suede Leather , Easy Spirit Women s Traveltime Mule Blanco / Verde oscuro / Cuero Multi , Drew Smiles Slip para mujer Negro / Microdot Zapatillas DC Mujeres Chelsea TX SE Sneaker Azul marino blanco Onitsuka Tiger Asics Unisex Tiger Corsair Aciano Azul / Azul Oscuro , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Blanco natural , Zapatillas de golf adidas W Climacool Knit para mujer Tiza Púrpura / Azul / Coral , Sandalia Vince Westcoast para mujer Vince Caucho magenta Plataforma CLARKS Leisa Grace para mujeres Blanco Sandalias Tabora Birkenstock para mujer Crystal Rose Birko Flor Sandalia Haviva Slide Dolce Vita para mujer Tela negra , Sandalia para mujer FRYE Ally Criss Cross Slide Negro , Calzado de vestir Genoveva para mujer Calvin Klein Estaño , Bomba Sadie Wedge 40MM de Cole Haan para mujer Fig Gamuza Zapatos de ciclismo para mujer Giro 2017 Petra VR Dirt Blanco Zapatillas Spike Cross Country Kilkenny XC5 Spike de Country Saucony Aqua / Vizi Coral Bella Vita Womens Mel-Italy Blush-gamuza , Botas de tacón de aguja de piel de oveja con detalles de piel de oveja puntiagudos de Themost Women s Feinty Toe Gris Botines cortos Farah de mujer SOREL Negro , Botín Matisse Fury Mujeres Punta Redonda Gris Tobillo marrón , Hush Puppies Women s Graham Blucher Oxford Cuero marrón

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 Calzado casual tejido NIKE NIKE para B077BF9MD8 tejido hombre Bronce Negro hombre Cumbre Blanca 700 33db14a 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

Thirteen elementary straightedge 682740
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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.