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,

Amortiguación 635v2 New Balance para mujer Plata Zapato deportivo ASICS Gel-Excite 4 para mujer Diva Azul / Blanco / Sol Zapatillas de running NIKE Air Max Flair para hombre Piedra Pómez / Legión Verde , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Gunsmoke / White-atmosphere Grey Zapatillas de running adidas Alphabounce HPC W para mujer Negro / Negro / Onix New Balance Women s w990v4 Running Shoe Armada Zapatillas de running NIKE Airstream Sequent 3 para mujer Negro / antracita , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Azul , Saucony Triumph ISO 3 Running Sneaker para mujer Azul marino Zapatillas de running ASICS Gel-Cumulus 19 para hombre Coralino / Azul oscuro / Azul oscuro Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Supercell / Sunrise , Camiseta adidas Swift Run W para mujer Blanco / Blanco / Blanco , Zapatillas de running adidas Alphabounce Beyond W, para mujer Gray Four / Carbon / Dark Solid Gray Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Platino Blanco / Blanco-puro , Zapatillas para correr Mizuno Wave Inspire 12 para mujer Royal Purple / Silver , Zapatillas NIKE Roshe LD-1000 QS para mujer Varsity Royal Varsity Maize White 401 , Zapatillas de running New Balance 990V3 para mujer Gris Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Trueno / Blanco , PUMA Men s Roma Basic Sneaker Blanco / Nueva Marina Pantalón largo clásico NIKE Golf para mujer Tech Essentials Gris oscuro / Platino puro / Platino metálico / Negro , Zapatillas de running W1540V2 New Balance para mujer Blanco azul , Zapatillas de running ASICS Gel-Contend 4 para mujer Blanco azul , Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Negro / Fantasma , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Gris Medio / Gris Medio / Rosa Concha Nike Men s Air Presto Essential Adoquines / Voltios / Antracita Zapatillas para correr Saucony Women s Cohesion 10 Gray Mint New Balance Vazee Running Shoe-Breathe Pack Zapatillas de deporte de moda para mujer Arrecife , Short NIKE Tempo para mujer Obsidiana oscura Zapatillas de running TOPO FLI-Lyte 2 para mujer Verde / Coral ,
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 OluKai Kulapa Kai Dark Java / Dark Java , Zapatillas adidas Outdoor Terrex Agravic Speed ​​Shoe de mujer Easy Coral, Negro, Blanco , Propet Women s Fisherman Sandal Mezclilla Skechers Women s Walk Walk Lite-Shanti Loafer Flat gris L Artiste de Spring Step - Sandalias estilo guante de cuero para mujer Gray Multi , Zapatillas de running AMPLICA para mujer ASICS Azul índigo / Azul índigo / Coral destello , Zapatillas de running ASICS GT-2000 6 para mujer Negro / Blanco / Carbono , Zapatillas para correr Saucony Freedom Iso Unisex Gris Zapatillas para caminar Skechers Women s Go 4-Brisk gris , Zapatillas de running ASICS Gel-Contend 4 para mujer Podar / Plata / Negro Zapatillas para caminar Go Step Lite-Solace de Skechers Performance para mujer Carbón JSport by Jambu Women s April Gladiator Sandal Armada , Zapatillas MBT Women s Speed ​​16 Slip On Walking Polvo Azul / Fucsia Sandalia de mujer de Steve Madden Oro rosa , Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Velcro Bristol Mujeres Mary Jane Zapatos atléticos Rosado , Zapatillas Skechers para mujer Go Walk Cool-15651 Sneaker Coral , Jordan Air 7 Retro BG Boys Sneakers 304774-034 - , Sam Edelman Women s Bay Slide Sandal Cuero metálico yute b.o.c Sandalias de cuero Bellisi para mujer Coffee Tooled , Así que Danca CH50 Character Shoe con 1.5 Heel Broncearse Reebok Zapato de baile Pilox 1.0 para mujer Colegial azul marino / blanco / veg bronceado 50 tonos de oro 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) 7004- Oro Sandalias Peep Toe de Misu para mujer Latin Salsa Blackgo Practicar zapatos de baile para salón con tacón de 2.75 \ Negro Salsa de baile femenina muy fina Zapatillas de baile de tango 1683 con tacón de 2.5 \ Cuero plateado Dr. Comfort Vigor Mujer diabético terapéutico Extra profundidad senderismo botas de cuero de encaje castaña , Botas Pearlie de Cole Haan para mujer Cuero Morel Bota de senderismo militar Pallabosse Hi Cuff Palladium para mujer Fallen Rock / Rainy Day , Jambu Women s Tangerine Oxford Flat Teal Zapatillas de deporte de mujer Easy Spirit de Easy Spirit Negro Propet Walkers Evolution Oxford lavable de las mujeres Gris claro / azul claro ,

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 NIKE Running Women s Free 5.0 Running Sneaker naranja s naranja 09a19cd 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.

Carretera en funcionamiento
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.