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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Sintético único
Eje mide aproximadamente parte baja del arco
Cuero Upper
Eastland Exclusivo activo Plantilla de espuma viscoelástica
CBC Suela de EVA
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

Castine Oxford B01N5I52AW de mujeres de Eastland Oxford Eastland Nuez moscada 50a7fc9

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,

TOMS Kids 10009918 Alpargata-K Arpillera metálica natural TOMS Womens Katalina Espadrille Sandal Cordón floral negro TOMS Wedge para mujer Stella Drizzle Grey Lurex Woven , Easy Spirit ON CUE Oxford para mujeres Negro Negro Merrell Solo Paloma Salvaje , Oxford de Drew Amora para mujeres Marrón-nubuck Zapatos holgazanes Mila de Trotters para mujer Cuero Nappa Suave Negro Finn Comfort Women s Ikebukuro Oxford Indianred Petka , Zapatillas de deporte Arizona Trotters para mujer Armada , Cobb Hill Rockport Mujeres Nadia Mary Jane Flat Negro , G.H. Azul Zapatillas de deporte lavables Walker Propet para mujer Sr Indigo , Timberland Women s Killington Oxford Blanco negro , Bota unisex New Rock M.1482X-S1 Negro , Rieker Womens L3249 Nikita 49 Brandy / Ozean / Schwarz , Qupid Women s Newbie-24 Oxford Poliuretano Slv Glitter , Zapatillas Loraty Easy Spirit para mujer Azul , Calzado para caminar CLARKS Women s Wave Andes Nubuck azul marino , Orgía de Kamryn resistente a la intemperie de Earth Orígenes de mujeres Negro CLARKS Zapatillas Tri Etch para mujer Cuero rosa nude Zapatillas de deporte Novo Demi de mujer K-Swiss Nomad Brown / Hawthorn Rose , Geox Kookean 4 Oxford para mujer Plata , Drew Shoe Women s Parade II Oxford Ternero negro Thorogood Women s Athletic Postal Oxfords Negro Mocasín de lona Minnetonka para mujer Lona azul marino School Issue Women s Varsity Blanco negro , Botín Hayes Gore de Cole Haan para mujer Ante de piedra de hierro SeaVees Women s 10/60 Buck Oxford Latón Zapatillas de deporte de mujer Alegria Essence Botanicool
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 deportiva Tirra para mujer Teva Simplemente Taupe Sandalia Teva W Terra Fi 4 para mujer Rocío Atlántico Norte , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Pure Platinum / Dark Grey , Chaco Women s ZX2 Classic Athletic Sandal Edredón Melocotón , Adidas Women s Adilette CF + Logo W Sandalia deslizante Ash Blue / Aero Blue / Black , Converse Chuck Taylor All Star Seasonal Color Hola Ratón , Sandalia NIKE Benassi JDI para mujer Blanco , PUMA Men s Roma Basic Sneaker Blanco / Peacoat / avena SAS Womens Sunny Pecan Brown Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Estaño Sandalia Saturno de mujer Miz Mooz Negro PUMA Men s Roma Basic Sneaker Blanco / Llama escarlata / Glaciar gris , Drew Shoe Women s Paradise II Velco Slip-on Ternero negro , Birkenstock Arizona Sandalia de cuero unisex Taupe Suede , Supra Vaider LC Sneaker Lana azul marino / plata / blanco Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Deep Jungle / White Clay Green , Sandalia para mujer FRYE Ally Criss Cross Slide Blanco Sandalia de cuña OAHU de China Laundry Women Blanco , BareTraps Women s Rebecca Slide Sandal Cameo Rose Sam Edelman Women s Slide Sandalia Denim Blue , Circus de Sam Edelman Sandalia de mujer Raquel Slide Plata blanda Zapato de Split Flex para mujeres de Bloch Dance Negro , Bota de media caña Georgia GB00065 marrón , Bota de montar de Donnily de Calvin Klein para mujer Shadow Gray , Pleaser - Botas para mujer, con tacón y punta abierta, 8 \ Blk Pat / Blk Trotters para mujer Dea Slingback Pump Negro Negro Bota de moda Dr. Martens Rakim Gris Rieker Antistress Mujeres 41730 Slip-on Muskat Cristallino / muskat Bogota Salvatore Ferragamo Womens Pisa 70 Suede de arcilla , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Ante negro

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.


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 Castine Oxford B01N5I52AW de mujeres de Eastland Oxford Eastland Nuez moscada 50a7fc9 it out, and I got a chance to try out this new graphics software.


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.

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

Geometry Construction Reference

Thirteen elementary straightedge 38511
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.