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

Pantalón largo Tech 19712 clásico NIKE Golf para Blanco mujer Tech Essentials Negro/ Blanco/ Gris Oscuro 08d1035

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.


Pantalón largo Tech 19712 clásico NIKE Golf para Blanco mujer Tech Essentials Negro/ Blanco/ Gris Oscuro 08d1035

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 de montaña de montaña Akyra La Sportiva para mujer Esmeralda / menta Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / oscuro, pasas de rosa, mortal , Zapatillas de deporte Nike Roshe Run para mujer Deep Granet / Black-gym Red-vry Berry NIKE Women s Tanjun Negro / Negro / Blanco New Balance Women s w990v4 Running Shoe Komen Pink , NIKE WMNS Zoom Fly Women 897821-411 Equator Blue / White , Zapatillas de running Mizuno Wave para mujer Wave Horizon Dark Shadow / Clownfish Zapatillas para correr Saucony Freedom Iso Unisex borgoña Zapatillas de running New Balance Urgev2 para mujer Silver Mink / Alpha Pink , Zapatillas de running NIKE Air Max Torch 4 para mujer Negro / Plata Metálico / Rosa Flash , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Menta / Plata / Azul , Zapatillas para correr Under Armour UA Speedform Fortis 2.1 para mujer Negro / Gris Glaciar / Gris Glaciar , Zapatillas de deporte con cordones Sneakers de hombre Carson Runner de PUMA Blue Atoll-white , NIKE Mujeres Downshifter 7 Negro / Mtlc Hematita Antracita Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Nube de lluvia / Artic Fox , Zapato de running ASICS GT-1000 6 para mujer Porcelana Azul / Humo Azul / Blanco , Zapatillas de running adidas Springblade Razor para mujer Power Teal / Frost Mint / Sol Gold Zapatillas de running NIKE Lunartempo 2 para mujer Rosado Zapatillas de running NIKE Airstream Sequent 3 para mujer Sequoia / Light Bone / Metallic Silver , Zapatillas de running adidas Mana Bounce 2 W Aramis para mujer Blanco / Metálico / Plateado / Negro , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Negro / Negro / Fantasma Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Deep Porcelain Blue / Overcast Zapatillas de running ASICS Gel-Contend 4 para mujer Plata / Campanula / Carbono Zapatillas de running Kinvara 5 Saucony para mujer Plata / Vizi Coral / Citron , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Niebla del océano / Cirrus Blue / College Navy , Zapatillas de running ZanteV2 Breathe de New Balance para mujer Blanco / Hola Lite Zapatillas de running Wstro para mujer New Balance Negro / gris / Lime Glow Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Baya del Saúco / Coral Vivo Zapatillas para correr Under Armour UA Speedform Fortis 2 para mujer Cubierto Gris / Blanco / Cristal
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 ASICS 1012A019 Zapatillas para correr GEL-Kayano 25 (2A) para mujer, color carbón / gris medio - 10 2A ES - NIKE Kids 5.0 gratis (grande) Rosado , NIKE Women s Tanjun Lava Glow / Blanco / Carmesí total , Sandalia KEEN Women s Venice H2 Beet Red / Neutral Gray , El corredor de trail impermeable Xa Pro 3D Cs para mujer de Salomon Sombra , Zapatillas de trail running WT10v1 Minimus para mujer New Balance Gris azul , Teva ALP Premier - Mujeres Armada , Yellow Box Mujeres Janalee Broncearse Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Plata CLARKS Mujer Gracelin Mara Flat Armada , New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Blanco Blanco , Entrenador cruzado Fresh Foam Crush V1 de New Balance para mujer Coral vivo / blanco Sandalias para mujer TOMS Lexie Open Toe Casual Slingback Lona / bordado azul marino , Zapatillas de deporte Fit Fitness GEL de ASICS para mujer Taupe / Cotton Candy / Coral Rose Hush Puppies Men s Gus Oxford Cuero negro ASICS Tiger Womens Gel-Lyte V Skyway / Skyway Zapatillas de golf Go Golf Blade de Skechers Performance para mujer Blanco / rosa Sandalia de la diapositiva de Benicia de las mujeres de Dolce Vita Ropa de Ceniza , Sandalia Mila Slide de mujer Naturalizer Ante negro CLARKS Leisa Cacti Slide Sandalia para mujer Cuero negro Marc Jacobs Women s Aurora Mule Plata , Sandalia de cuero Nirveli de Nine West para mujer Cuero negro Bomba Sadie Wedge 40MM de Cole Haan para mujer Rockridge Suede , Propet Cush N Foot Slipper & Oxy Cleaner Bundle Armada Bota de mujer Vince Camuto Estiva Cabernet TOMS Women Laurel Burnished Suede Boot Gamuza bruñida anfora , Bombardero de mocasines de conducción Twisted X para mujer / Morado - Calzado de exterior retro inspirado Zapatillas de fútbol adidas Performance Women s Ace 17.4 FG W Tinta Negro / Fácil Coral / Misterio Bota Darcy para mujer Easy Street Negro / Negro Cocodrilo / Gore , CLARKS Women s CloudSteppers Sillian Jetay Flat Tela roja jaspeada

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.

Pantalón largo Tech 19712 clásico NIKE Golf para Blanco mujer Tech Essentials Negro/ Blanco/ Gris Oscuro 08d1035 - movie-store.online


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 Pantalón largo Tech 19712 clásico NIKE Golf para Blanco mujer Tech Essentials Negro/ Blanco/ Gris Oscuro 08d1035 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

Thirteen elementary straightedge 172269
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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.


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