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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Mango de plástico para Control de torsión
Incorporado StepSaver Orthotics
Térmico Agarre de goma (TR) Suelas
Cuero Parte superior
Ajustable Correa del talón, elástico Sangre
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,

TOMS - Slip-On de lienzo clásico para hombre Negro / Ciruela Trotters Lucy Ballet para mujer Negro Bomba Sadie Wedge 40MM de Cole Haan para mujer Fig Gamuza Naturalizer Michelle s Dress Dress Pump Negro brillante , Glerups modelo unisex BR Slipper Gris , Zapatillas de mujer Foamtreads Coddels Armada Botín de plumas abajo de las mujeres de Cabiniste Delta azul / gris , Haflinger zigzag unisex Gris , Haflinger zigzag unisex Capitanes azules , Foamtreads Barry Masculina Negro , Jacques Levine Bel Esprit Pom Pom Gris pardo Vionic Womens Kari Cable gris , Bomba Parigi de LifeStride para mujer Black-microfiber Zapatos OLUKAI Pehuea - Mujer Teal / Teal , Magic-MC New Summer Zapatillas Cork Slides Pisos con 2018 Nuevas mujeres ocasionales Deslizamiento en toboganes de playa Zapatillas Flip Flops Talla extra Lugar , Zapatilla Nohea de OLUKAI - Mujer Trench Blue / Trench Blue Bomba Hazel para mujeres Sam Edelman Midnight Navy Sequin , Calvin Klein Women s Gabrianna Pump Limonata , Zapatilla de sombra de mujer KENDALL + KYLIE Gris , Bota Deia para mujer de TOMS Cuero negro Zapatilla de pies Cush N de Prop & eacute; t para mujeres Corduroy negro TOMS Clásicos de mujer Lona azul marino Propet Cush N Foot Slipper & Oxy Cleaner Bundle Merlot Corduroy Bomba Francesca de mujer Adrianna Papell Satén negro Pantuflas Mahabis Classic para hombre gris Gris oscuro / Larvik gris claro TOMS Clásicos de mujer Negro sobre negro Zapatos de tacón alto con tachuelas y tachuelas Kaitlyn Pan en punta Rojo mate / rojo Trim / Red Studs Calzado de vestir Genoveva para mujer Calvin Klein Blanco suave Trotters Lucy Ballet para mujer Desnudo
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 tenis ASICS Gel-Game 6 para mujer Plata blanca Zapatillas de running Reebok Print 2.0 para mujer Negro / azul polar / peltre Zapatillas de running adidas Adistar Boost ESM para mujer Gris claro / verde lima Vionic Rest Farra - Sandalias de apoyo para mujeres Patente roja , Naughty Monkey Mujer Hey Pony Sandal Leopardo Zapato para caminar Rae Ryka para mujer Azul , Sandalia de mujer de nivel medio Aerosoles Denim Combo Zapatillas de running Brooks Womens Glycerin 15 Neutral Max Cushion Negro / rosa pavo real / ciruela Caspia , Bomba Nikole para mujer Via Spiga Gamuza de visón Zapatilla Ascot de hombre UGG Bubbly / Sahara , Calzado de entrenamiento casual Comfort New para mujeres wx857v2 de New Balance Negro Zapato Ryka INFLUENCE2.5 Cross-Trainer para mujer Negro / púrpura Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante negro , Skechers para el trabajo de la mujer Albie Walking Shoe Blanco , Calzado de senderismo Go Joy 15602 de Skechers para mujer Armada , Zapatillas de skate Etnies Women s Scout W s Negro / Blanco / Borgoña Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker azul Zapatillas de golf Sunnylite V2 para mujer de PUMA Blanco / azul frío / gris violeta Charles by Charles David Sandalia deslizante Goldie para mujer Oro rosa , Sandalia Aerosoles Midday Women Slide Tela negra Sandalia Corey de Dolce Vita para mujer Multi Raffia Natural Sandalia Dolce Vita Celita para mujer Cuero negro , Sandalia plana deslizante trenzada metálica Soludos para mujer Nude / Pale Gold Calvin Klein Women s Gabrianna Pump Platino blanco , Zapatillas de verano mahabis Agen Aqua / Skane Yellow Zapatillas Spike Ballista Saucony para mujer Azul rosado Trotters Jake para mujer plana Negro Splendid Midi Perf Vionic para mujer Slip-On Rosa empolvado CLARKS Mujer Ashland Bubble Slip-On Cuero marrón Trotters Liz Loafer para mujer Hueso

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 Sandalia 19352 de de golf Sandbaggers Carrie para mujer rojo rojo d4f0f5b 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 2270165
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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.