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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Importado
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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

Sandalia Terrell Terrell Ivy Wrap 10368 Sport Monumento Merrell para mujer Monumento 7014ec6

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,

Chaco Women s Z1 Classic Athletic Sandal Gaulteria CLARKS Sandalia de pescador de melón Tuvia para mujer Broncearse , OluKai Ohana Dalia / Negro Sandalia deportiva Kankitta de Vince Camuto para mujer Cloudy Gray Combo KEEN Hombres Newport H2 Sandal Bossa Nova / Gárgola Sandalia Uneek O2-w de mujer KEEN Negro Negro Sandalia deportiva Nolita para mujer Anne Klein Tela negra , Encaje de quimera de Columbia para mujer Zing / Super Sonic , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Taupe liso Sandalia deportiva Kankitta de Vince Camuto para mujer Summer Cognac Combo , Sandalias Newport H2 de KEEN para mujer Cáñamo / azul provincial Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Negro Sandalias Newport H2 de KEEN para mujer Cáñamo / Gorra blanca Fabulicious Women s Caress 401Dm sandalias de diapositivas Clear / Silver Multi Rhinestone Sandalia Kea Vent mujer Columbia Marrón oscuro, Hawk Sandalia Drew Angela para mujer Beige-multi , Sandalia de susurro de mujer Keen Sombra Oscura / Cerámica Drew Womens Cascade Nubuck negro , Xero Shoes Z-Trail Sandalia ligera - Descalzo-Inspirado Ruta de senderismo, Correr sandalias deportivas - De las mujeres Carbón Negro / Mocha Tierra Sandalias de encaje Ros Hommerson para mujer Black Snake Print Sandalia de susurro de mujer Keen Imán / Sangría Sandalia Drew Element para mujer Nubuck azul marino , Sandalia deportiva Zvolv X de Chaco para mujer Alloy Dancer , Chaco Mujer Zcloud X2 Sandalia Deportiva Runa Teal , Sandalias KEEN Mujer Ana Cortez Flip-W Negro Sandalia Savannah de Ryka para mujer Beet Red / Hyper Pink , Chaco Women s ZX2 Classic Athletic Sandal Credo Golden Sandalias de encaje Ros Hommerson para mujer Microtouch negro Chaco Mujer Diana Sport Sandalia Avo florentino ,
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 Gobiv2 para mujer New Balance Semillas con Grove Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro , Zapatillas de deporte Onitsuka Tiger Mexico 66 Gris claro / azul marino , Skechers Performance On The Go 400 Tropical Flip Flop gris , Zapatillas de running Skechers GOrun Forza 2 para mujer Carbón de leña / Púrpura Calzado de running New Balance para mujer 530v2 de New Balance Ártico Fox / amanecer blanqueado Sandalia deportiva Chaco Zvolv 2 para mujer Swell Nickel , Sandalia Yucatan ECCO Mujer Atmósfera / Hielo Blanco / Negro Zapatillas de running New Balance 990V3 para mujer Gris / rosa , Zapato deportivo artesanal de mujer superfeet Addy Yuca Reebok Zapatillas para caminar de estilo libre para mujer Blanco / goma CLARKS Sandalia de vestir Leisa Gianna para mujer Cuero azul marino , Sandalia Wavey para mujer OTBT Marrón medio Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Púrpura / plateado Vans Unisex Brigata Zapatillas de skate, náuticas Zapato de barco inspirado Clásico, Cómodo y duradero en Waffle original Suela Monjes Robe , Orthofeet 877 Women s Comfort Diabetic Therapeutic Extra Profundidad Calzado Lycra Velcro , adidas Originals Mujer Superstar Blanco / Lino Verde Hielo Tela Púrpura Zapatillas de deporte Chimera Mesh Sneaker de Columbia Dark Mirage, Júpiter Clásicos de base Slip-on (Tm) para hombres de Vans Negro Negro , Zapatillas de golf Ignite Blaze Sport Disc PUMA para mujer Gris Violeta / Blanco , Sam Edelman Sandalia Bernice Slide para mujer Saddle Vaquero Saddle Leather , Sandalias con cordones Shellie de Kate Spade New York para mujer Medianoche , 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 2829- Oro Sparkle & Snake Zapatillas Dr. Mary Comfort Flauta Lycra Elasticables Diamante Mary Jane Licra , Baabuk Wool Sneaker - Mujer Azul claro / amarillo Zapatos planos de tacón bajo YDN para mujer con tacón bajo, deslizamiento en zapatillas Bowknot Zapatos antideslizantes de verano en deslizamiento Turquesa Calvin Klein Women s Gabrianna Pump Coñac Sandalias Nuria de Toni Pons para mujer Ante marrón , Zapatillas Nike Women s Flex Trainer 5 Antracita / Ghost Green / Chlorine Blue Predicciones Comfort Plus Women s Karmen Pump Patente negra ,

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.

Sandalia Terrell Terrell Ivy Wrap 10368 Sport Monumento Merrell para mujer Monumento 7014ec6 - movie-store.online


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 Sandalia Terrell Terrell Ivy Wrap 10368 Sport Monumento Merrell para mujer Monumento 7014ec6 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 2838079
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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.


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