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,

Zapato de running 840v4 New Balance para mujer Gris oscuro Zapatillas de running New Balance Boracay V3 para mujer Blanco negro Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Sunset Pulse Arctic Punch Wht ONEMIX Zapatillas de deporte de aire para mujer Coloridas reflexiones Zapatillas de deporte de estilo deportivo Rosado Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / gris oscuro , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Mango brillante negro / blanco , Zapato de running Saucony Zealot ISO 2 para mujer Té / Algodón , Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Supercell / Sunrise Zapatillas de deporte Reebok Print Run Dist para hombre Cool Shadow / White , Nike Hombres Air Zoom Pegasus 33 Lobo gris / negro / gris oscuro / azul Zapatillas de running ASICS Gel-Nimbus 18 para mujer Melocotón Melba / rosa fuerte / guayaba , Zapatillas de running New Balance 1340v3 para mujer Blanco / morado , Zapatillas de running Altra AFW1845F Provision 3.5 para mujeres Negro NIKE Roshe G Spikeless Golf Shoes 2018 Mujeres Blanco negro , Zapatillas de running Saucony para mujer Stretch N Go Breeze Negro , Zapatillas de running NIKE Lunartempo 2 para mujer Chalk Blue / Black / Racer Blue / Hyper Orange Zapatilla Skechers para mujer Go Run 600-Obtain Frambuesa Zapatillas de running AVIA Avi-Pulse para mujer Negro / Gris hierro / Rosa Athena , Zapatillas de running Saucony Women s Freedom ISO Berry / rosa , Zapatillas de running adidas Cloudfoam Ultimate para mujer Blanco / Blanco / Gris Tres Zapatilla de running AS-Gel-Kayano 24 Lite-Show para mujer ASICS Índigo Azul / Negro / Reflectante , Zapatillas de running New Balance 990v4 para mujer Faded Rose / Komen Pink , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Dragon Fruit / Overcast , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Gris claro Zapatillas de running Saucony Women s Breakthru 3 Azul / menta Zapatillas de running 430v1 New Balance para mujer Gunmetal / Seafoam , Calzado deportivo New Balance Vazee Urge v1 para mujer Blanco negro , Calzado casual NIKE Flyknit AF1 para mujer Black / Black-team Red-clr Jade Zapatillas de entrenamiento Nike Air Zoom Fearless Flyknit para mujer Black / White-Volt ,
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 NIKE Men s Roshe Two Running Shoe Gimnasio Rojo / Negro / Vela / Voltio , Zapato deportivo ASICS Gel-Kayano 24 para mujer Negro / Phantom / Blanco OluKai Ohana Jalea de guayaba NIKE Women s Tanjun Negro / Negro-antracita-blanco , Zapatillas de trail running New Balance WT00 Minimus Zero v2 para mujer New Balance Púrpura / rosa , Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Velcro Bristol Mujeres Mary Jane Zapatos atléticos Blanco XYD Women Open Toe Low Heel Strappy Rhinestone Sandalias Correa de tobillo vestido de novia Bombas con cremallera Serpiente Imprimir Zapatillas deportivas Easy Spirit Punter para mujer Blanco / gris claro Zapatillas para mujer Cloud 2568 Caliber Slide Plata , Zapatillas de deporte Skechers Sport Women s Empire Inside Look Negro / Multi jiandick Rhinestone de las mujeres del tobillo Bootie Prom Sandalias de tacón de noche Vestido de noche Stiletto High Heel Negro-plata , ASICS Gel-Fit Sana 3 Cross Trainer para mujer Invierno Bloom / Plata / Violeta Zapatillas Slip-On Skate Print de Vans para hombre Oro / Blanc De Blanc Zapatillas de running Golden Spike de Altra para hombre Negro / Rojo , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Rosado Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Blanco negro , Sandalia Dolce Vita Delana Slide para mujer Lino Indigo , Sandalia Cato Dolce Vita para mujer Lt Blue Denim , Zapatos OLUKAI Pehuea - Mujer Trench Blue / Trench Blue , Zapato de campo traviesa Kilkenny XC5 para mujer Saucony Teal | , Sandalia Birkenstock de cuero Florida para mujer Negro Birko-flor Cobb Hill Rockport Women s Parker CH Flat Lino , Bota de tobillo Rexford para mujer Cuero negro de grano completo Dr. Martens Womens 1461 Virginia desnuda K-Swiss Womens Hoke SNB CMF Gota de limón / Nubuck blanco Rockport Cobb Hill Collection Mujeres Cobb Hill Sharleen Pump Cuero negro , Bota Cobb Hill para mujer Penfield Cuero negro , Red Wing Heritage - Bota Moc-W para mujer, 6 \ Legado de Oro , Hush Puppies Womens Cesky Perf Oxford Cuero blanco Zapatillas Slip-On Roamer SAS para Mujer Mezclilla ,

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 ASICS Kanmei Shoe Casual para gris mujer Gris B077MRPFTR 15278 medio/ gris medio/ carbono fe5d61e 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 1038808
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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.