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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Textil y sintético
Importado
Caucho único
Eje mide aproximadamente Tapa baja del arco
Sin costura Construcción: El uso de inconsútil materiales reduce el potencial para irritación y fricción causada por tradicional puntadas y costuras.
Discreto Ojetes: Independientemente ojales colocados desechar la tensión del encaje creando un personalizado entorno apropiado y superior mejorado comodidad.
extraíble Plantilla: un calcetín revestimiento que puede ser eliminado para acomodar una ortesis médica.
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

Zapato de running Medio Fuzex B01NBJPUJW Fuzex Rush para mujer de ASICS Medio gris/ Bay/ Flash Coral 208156e

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,

Calzado deportivo New Balance Vazee Urge v1 para mujer Púrpura / amarillo Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print , Zapatillas de entrenamiento Nike Womens Metcon 3 Cool Grey / Solar Red-black-pure Platinum Zapatillas de running NIKE Air Huarache para hombre Blanco / Blanco-blanco-negro New Balance Women s w940v3 Running Shoe Plata Zapatillas de running NIKE Women s Air Max 1 Ultra Essentials Azul , nike womens air max 1 JCRD zapatillas para correr 819808 zapatillas zapatos Hematita Metálica / Negro-gris oscuro , Zapatillas de running ASICS Gel-Cumulus 19 para mujer Negro Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Negro Zapatillas de running adidas Women s Edge Lux W Gris / gris / blanco cristal Zapatillas para correr Mizuno Women s Wave Enigma 6 Púrpura / blanco Zapatillas de running acolchadas 420v4 para mujer New Balance Gris , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Tierra púrpura / negro , Zapatillas de running adidas Alphabounce Beyond para mujer Blanco / Gris / Gris , Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Azul , Saucony Women s Cohesion 8 Running Shoe Zapato para correr ASICS Conviction X 2 para mujer Negro / Carbón / Flash Coral Zapatillas de running adidas Cloudfoam Xpression W para mujer Blanco / Blanco / Haze Coral Zapatillas para correr Brooks Cascadia 12 para mujer Peacoat / Flor de la pasión / Bluebird Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Negro / Gris oscuro / Azul Racer / Negro , Zapatillas de running New Balance para mujer 940v3 Azul negro , Zapatillas de running NIKE Air Huarache Run para mujer Dark Stucco / Pale Gray Brooks Women s Adrenaline GTS 17 Plata / púrpura Cactus flor / Bluebird , Zapatillas para correr New Balance W840V3 para mujer Gotita , Zapatillas de running ASICS Gel-Cumulus 19 para mujer Aruba azul / azul de Aruba / azulejo turco , Nike Hombres Air Zoom Pegasus 33 Gris oscuro / negro - blanco NIKE W Air Huarache City Low Women Ah6804-100 Blanco / Blanco-blanco Brooks Womens Ghost 10 GTX Negro / Azul pavo real / Malvavisco Nike Men s Air Presto Essential Oro negro / negro metálico
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 enrejado Sirah para mujer Merrell Negro Zapatillas de running NIKE Air Huarache para hombre Washed Teal / Pure Platinum , Adidas Women s Adilette CF + Logo W Sandalia deslizante Shock Pink / Collegiate Navy / Aero Pink Zapato de running ASICS GT-1000 6 para mujer Insignia Azul / Plateado / Rojo Rouge , Trotters Mujer Simmy Mary Jane Flat Ante azul marino , Zapato de trabajo con cordones Skechers for Work Synergy Sandlot con punta de mujer Negro , Zapatillas deportivas Under Armour Xc Brigade Spike Negro (001) / Blanco Prop & eacute; t Zapatillas para caminar TravelFit para mujer Lima / Azul , Calzado de entrenamiento casual Comfort WX623v3 New Balance para mujer Plata blanca Nike Tech Xtreme Cadet Glove Amarillo de voltaje negro / blanco Easy Spirit - Zapatillas Faisal2 para mujer gris Prop & eacute; t Zapatillas para caminar TravelFit para mujer Naranja / Rosa Diadora N9000 MM Bright Verde Supra Skytop Skate Shoe Blanco negro Golf de mujeres y rsquo; s Peso ligero genuino Cuero hecho en Calle de Brasil Mott Rendimiento de golf Marc Joseph NY Zapatos de moda Patente de la marina de guerra , Sandalias con tirantes Rosetta-1 de mujer Pierre Dumas Combo de la Marina Sandalia de cuero Nirveli de Nine West para mujer Cuero negro Sandalia Otto Slide de mujer Yellow Box Sonrojo Sandalia de raso de satén Feeltheluv de mujer de las nueve oeste Satén negro Grishko 0509/2 2007 Pro-Flex , Very Fine Dance Shoes 2707 (Grado de competición) 2.5 \ 2707- Satén marrón Zapatilla Dakota de las mujeres de Australia UGG Chemise Pink , Salón de baile de mujeres Zapatos de baile Tango Zapatos de boda de la salsa 9624EB Cómodo-Muy Fino 2.5 \ de 5] Cuero negro Zapatillas de ciclismo para mujer Sidi Alba Road Bicycle Negro , SOREL Bota de cuero baja grande para mujer Tabaco Bota del oeste de Denver de Dan Post para hombres Negro Sobre de mujer Aerosoles A2 Tela azul marino , Zapatilla de cuña para mujer Jacques Levine # 4640 Metálico Multi T-JULY Womens Ladies Plataforma de cuña Sandalias de tacón bajo Diamantes de lujo Diapositivas Moda Deslizamiento en elegante Pantuflas inferiores gruesas Como la imagen , Bomba de vestido caliente de mujer Rose Petals Gold Micro ,

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 Zapato de running Medio Fuzex B01NBJPUJW Fuzex Rush para mujer de ASICS Medio gris/ Bay/ Flash Coral 208156e 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

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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.