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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tejido y sintético
3 Manera de usar, fácil de desmontar y combinar a través de cordón, perfecto para la luz urbana deportes
Exterior: microfibra sandalias roma alta con diseño tejido, ligero y flexible
Interior: 3D calcetín como calcetín zapatos interiores con Suela tachonada KPU, adecuada para caminar, conducir, hacer yoga u otro interior ejercicio, o como zapatilla de casa de ocio
Ecellon, transpirable y plantilla de espuma blanda
No slip sintético suela exterior, moldeada banda de rodadura, peso ligero, flexible
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,

Salomon Sense Ride Running Shoe - Mujer SANGRIA L398486 Salomon Women s Speedcross 4 CS W Trail Runner Sombra Zapatillas de running EM Trail M2 v3 para mujer Pearl iZUMi Perla ahumada / rosa brillante Salomon Women s Speedcross 4 W Trail Runner Dark Purple / White / Deep Lake Topo MT-2 Trail Running Shoes & Workout Headband Bundle Negro / Hielo Zapatillas de trail running Olympus 2 de Altra para mujer Plata / Verde Zapatillas adidas Outdoor Terrex Agravic para mujer Gray Three / Gray Four / Chalk Coral SCARPA Women s Spin Wmn Running Shoe Trail Runner Scuba Blue / Lemon , Topo MT-2 Trail Running Shoes & Workout Headband Bundle Negro / Hielo , Zapatilla de deporte Trail running wt690 para mujer de New Balance Teal / rosa Zapatillas de running Icebug Zeal2 RB9X para mujer Dahlia / Grape , Salomon Women s Speedcross 4 W Trail Runner Azul / azufre Zapatillas de trail running WT10v1 Minimus para mujer New Balance Gris azul Reebok Women s Ridgerider 2.0 Trail Runner Piedra de hierro / Mar Gris / Carbón / Calavera Gris / Negro / Plata / Peltre La Sportiva Ultra Raptor Mountain Running Shoe - Mujer Esmeralda / menta Zapatillas de running Gore-TEX 910v4 New Balance para mujer Pigmento / Porcelana azul , Zapatillas de trail running Salomon Women s Sense Mantra 2 W Nectarina / Aluminio / Sensación de Naranja Rookite 325 Trail Runner de mujeres Inov-8 Gris oscuro , Zapatillas de trail running Salomon Speedcross 4 para hombre Poseidón , Zapatillas de running Trail Trail Nomad para mujer Saucony Azul marino Vasque Men & fnof; _Ts Constant Velocity II Trail Running Shoes Vestido Blues / Neutral Gray , Zapatillas de trail running XA Pro 3D W para mujer Salomon Living Coral / Black Virtual , Camiseta sin mangas con espalda descubierta para mujer NIKE Breathe Blanco Columbia Montrail Women s Fluidflex X.S.R. Phantom Purple, Nappa Green Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Trueno Salomon Women s XA Enduro W Trail Runner Cielo nocturno , Zapatillas de running Salomon Women s Speedcross Pro 2 W Trail Negro Salomon Sense Ride Running Shoe - Hombre Azul marino Blazer / Bright Marigold / Ombre Blue , Zapatillas de trail running New Foam Gobi Neutral New Balance para mujer Guayaba / Negro ,
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 Sandalia Drew Angela para mujer Bone Smooth Leather Wolky Comfort Jewel Black Croco , Zapato NIKE Roshe One femenino Concord / Ember Glow-sail Sandalia Savannah de Ryka para mujer Gris / Purpl Zapatillas para correr New Balance W840V3 para mujer Gotita Sandalia de deslizamiento ultra Comfort NIKE para mujer Obsidiana / azul de cloro , La Sportiva Ultra Raptor Mountain Running Shoe - Mujer Baya Calzado de natación paso-naval Go Skechers Performance para mujer gris Skechers Performance Women s Go Run 400 Obstruct Negro , Calzado deportivo G-Defy Gamma-Ray para mujer Gravity Defyer Azul blanco , Zapatillas para caminar Ryka Women s Devo Plus 2 Negro / rosa CLARKS Sandalia plana Brizo Sammie para mujer Microfibra perforada de arena , CLARKS Womens Quartz Sandalia Arena Tierra Dalia Azul Sandalia de la correa de Sunvue de las mujeres alrededor de la ciudad de Merrell Arenisca Sandalia Saturno de mujer Miz Mooz glaciar , Zapatillas para caminar ProGrid Integrity ST2 para mujer Saucony Gris oscuro Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Puma White-puma Black , Pico Negro Zapatilla deportiva con cordones Slip-On Onitsuka Tiger Mexico 66 Ahumado azul claro , APL: Athletic Propulsion Labs Zapatillas de running mujer Techloom Phantom Deep Emerald / Opaque Melocotón , Zapatos de golf AsJei para mujer FootJoy Negro , Zapato de danza de gema Soleil moderno para mujer de PUMA Puma Black / Puma Black , Bota de vaquero Western Sterling Western de Ariat para mujer Naturalmente turquesa Crocs Unisex Ralen forrado Zueco Espresso / Espresso , Bota de invierno UGG Classic Mini II para mujer Niebla de Lavanda Bota de media caña Georgia GB00105 Marron oscuro Easy Spirit Cacia Mary Jane para mujer Gris SOREL Womens Joan Rain Wedge Chelsea Grafito / Negro , Zapato femenino Keli de Alegria Waverly ,

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 MT-onemix rosado Athletic Women de Men Awesome 3-en-1 zapatos ligeros Athletic Sneaker de Roma Oro rosado 8eac59a 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.

Trail Running
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1532759
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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.