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,

Mocasines con suela de goma con cordones de cuero Twisted X para mujer - Bombardero / Rosa Bomber / Neon Yellow Botín de tobillo Cyra Catelyn de Hush Puppies para mujer Mariposa , Botín de tobillo de Dallas para mujeres de Naturalizer Grano de café Jeffrey Campbell Womens Lilith-2 Boot Plaid Multi Botines Wander de mujer de NAOT Brushed Seal Brown Suede , Bota de ante FRYE Sacha Moto Shootie para mujer Gris oscuro , Botines de tacón de aguja de mujer Sammitop 8 cm Zip Up Botas cortas de invierno con piel Patente negra , Wolky Comfort Sneakers Ewood Cuero de terciopelo bordo FRYE Bota de cremallera doble Natalie para mujer Grano completo suave pulido negro Serie termal de Blundstone Negro , Bota de plataforma Ranger-301 para mujer Demonia Pleaser Black Pu , Vince Camuto Parka Women Blue Botines Taupe de medianoche Bota premium impermeable Timberland para mujer de 6 \ Susan G. Komen Ibis Rose Nubuck , Bota sobre la rodilla Franco Sarto para mujer Korinne Turba Bota de mujer Bearpaw Marlowe Aceituna , Justin Boots MSL504 Mujeres punta redonda de cuero marrón botas de lona Chocolate Barocco , Wolky Comfort Sneakers Ewood 40210 Cuero Antracita , Bota de tobillo Palladium Pampa Hi Orginale Red-604 Cocos de Matisse Women s Gaucho Boot Turquesa / Tostado Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro , Aquatalia Womens Fabienne Castaño de gamuza Botas para mujer InterestPrint Botas con cordones con diseño exclusivo Multi 3 , Botín de tobillo Cyra Catelyn de Hush Puppies para mujer Gray Shimmer , FRYE Mujeres Brooke corto interior Zip Tobillo Bootie Coñac NEWROCK New Rock Newrock Botas unisex de cuero negro metálico impresionante Biker - M.391-S1 Negro Calcetines de tacón alto para mujer, botines de media pierna con cordones, botines de tacón medio, botines de tacón de ante elástico para mujeres Brown Suede-6.5cm , Zapato de trabajo Rockport Work Mask Rk6465 para hombre Negro Tela Nine West Tria para mujer Negro / Multi , Botines de tacón de goma Medina III de Sorel para mujer Negro / Hyper Blue ,
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 Zapatillas de running Saucony Women s Breakthru 2 Crepúsculo / Oxígeno / Citron Sandalia deportiva W Sirra de mujer Teva para mujer Boysenberry Zapatillas de running Topo Athletic Ultrafly 2 - Mujer Hielo / Rojo , En la ejecución de calzado de mujer en la nube Blanco negro Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Gray / Racer Pink / Gunsmoke Calzado de entrenamiento cruzado TR 6 NIKE para mujeres en temporada Gris oscuro / Metallic Platinum / Fire Pink CrossKix 2.0 Athletic Water Shoes - Hombres y mujeres atlántico Zapatillas de running Nike Kaishi para hombre rojo , Inov-8 Mujer F-Lite 260 (W) Cross Trainer Gris negro Sandalia Marevick para mujer OTBT Nueva Taupe Zapatillas de running NIKE Airstream Sequent 3 para mujer Negro / antracita , Sandalia de mujer de nivel medio Aerosoles Denim Combo CLARKS Womens Un Reisel Myra Slide Sandal Cuero azul / gris SOREL Womens After Hours Slingback V-camel Brown Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Negro , Skechers Women s Go Walk Revolution Ultra Sneaker Carbón de leña / azul , Sandalias Birkenstock Unisex Arizona Marrón Amalfi de Cuero - 36 N EU / 5-5.5 2A (N) US - , BareTraps Women s Jaylyn Slide Sandal Negro / Plata , Sandalia Kai Dolce Vita para mujer Cuero Ivory Sandalia Cato Dolce Vita para mujer Pelo de ternero leopardo Mizuno 9-Spike Swift 3 Switch Softball Cleat Real / Blanco TOMS Clásicos de mujer Negro Negro , Zapatilla de cuña para mujer Jacques Levine # 4640 Metálico Multi , Zapatilla de deporte Stella Nd58 de El Naturalista para mujer Madera , Mulz Unisex House Shoes - Zapatillas interiores y exteriores de lana - Made in Europe Fuchsia Multi Easy Spirit ON CUE Oxford para mujeres Ceniza , Zapatillas de deporte de mujer Easy Spirit de Easy Spirit Gris oscuro , Rockport Mujeres Cobb Hill Lizzie Ballet Flat rojo , FLY London Womens Yomi765Fly Reef Cupido TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Tela negra lavada

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 Twisted X - Boy Turquesa Buckaroo Cowboy Boot Square Toe 11270 Cowboy - Ybk0005 Coñac 243a43e 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.

Botas
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.