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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Caucho
Textil superior es flexible y transpirable.
Inyectado suela de unidad (IU) proporciona peso ligero amortiguación.
Entresuela es lo suficientemente durable para servir como un suela.
Waffle suela tiene flex surcos para un peso ligero, sensación 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,

Zapatillas para correr NIKE Women s Air Zoom Structure 21 Armería Azul / Armería Azul marino-cirro Azul , Calzado de running casual para mujer New Balance WL555, para mujeres Negro , Zapatillas para correr Under Armour UA Speedform Gemini 2.1 para mujer Tahitian Teal / White / Lime Light , Zapatillas de running Under Armour para mujer Toccoa Grava (100) / Pulso Neo , Zapatillas de running ASICS Gel-Kayano 23 para mujer Blanco / Plata / Acuario New Balance Women s 555 Lifestyle Fashion Sneaker-Suede / Mesh Zapatillas para correr Gris / gris Zapatillas de running adidas Ultraboost X Parley para mujer Carbono, Carbono S, Azul Espíritu S Zapatillas de running NIKE Air Zoom Vomero 13 para mujer Piedra pómez clara / Resplandor verde Zapatillas de running adidas Women s Edge Rc W Negro / Plata Metálico / Claqua Zapatillas de running Classic-M Vibram para hombre Negro Zapatillas deportivas NIKE Lunar Apparent para hombre Negro rojo antracita blanco , Zapatillas de running NIKE Women s Air Zoom Structure 20 Negro / Blanco-cool Gris-puro Platino Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Plata / rosa , Inov-8 Mujer F-Lite 260 (W) Cross Trainer Púrpura / negro , Zapatillas deportivas NIKE Lunar Apparent para hombre Jade claro de carbono / negro claro , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul / Coral vivo Zapatillas de running Under Armour Charged Bandit 3 para mujer Negro (003) / Blanco Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Pure Platinum / Plata Metálica Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Blue Vapor / Camp Smoke NIKE Men s Benassi Just Do It Athletic Sandal Blanco / Real Academia Zapatillas de running Saucony Women s Jazz Low Pro Gris blanco , Brooks Womens Adrenaline GTS 18 Ébano / plata / rosa , Zapato de running de punto fuzeX para mujer ASICS Flash Coral / Negro / Amarillo de seguridad Zapatillas de running NIKE Air Huarache para hombre Washed Teal / Pure Platinum , NIKE Huarache Run Ultra Fashion Sneaker para mujer Cool Gray / Anthracite-black-white Zapatillas de running adidas Alphabounce Em W para mujer Amarillo hielo / Semi Solar amarillo / blanco PUMA Men s Roma Basic Sneaker Blanco / Rio Red / Steel Grey , Zapatillas de running ZIGOR para mujer Zapatillas de mujer Flyknit Epic React gris , Zapato deportivo ASICS Gel-Kayano 24 para mujer Blanco / azul violeta / amarillo de seguridad ,
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 Lanzamiento de Brooks Womens 4 Negro / Fucsia Púrpura / Azul deslumbrante Sandalias Terrell Ivy Lattice Sport de Merrell para mujer Negro , Saucony Women s Ride 10 zapatillas para correr Gray Berry Zapatillas de running HOKA ONE ONE Tracer para mujer - SS17 Púrpura , Zapatillas de running para correr adidas outdoor Kanadia 8 TR para mujer Negro / Core Pink / Trace Gray , Zapatillas de senderismo acuáticas para hombre Astral TR1 Junction Dirt Brown , Zapatillas de running New Balance 990V3 para mujer Blanco azul NIKE Women s Revolution 4 Running Shoe Ambiente Gris / Barely Gray-vast Gray , Sandalia Ahnu Women s W Serena Cork Negro Sam Edelman Womens Barry Cuero metalizado pulido de oro fundido oscuro , ASICS para mujer Gel-Fit Yui 2 SE Negro / Limelight / Plata , Orthofeet Probado Alivio del dolor Comodidad Serene Womens Orthopaedic Artritis extra ancha Diabetic T-Strap Zapatos Negro Zapatillas Slip-on de mujer Skechers You Inspire Negro Diadora Unisex Intrepid NYL Blue Estate , Zapatillas Superga 2750 Cotu para mujer Ecru Nike Tech Xtreme Cadet Glove Negro / Hyper Crimson / Hot Punch / Volt , Merrell Women s Mqm Flex Mid Waterproof Gris negro , Zapatillas de deporte Unisex Skate Trainers de Vans Authentic Pewter / Black Zapatillas de golf Ignite Blaze Sport Disc PUMA para mujer Gris Violeta / Blanco , Sandalia de la diapositiva Indi Kate Spade New York para mujer Neopreno rosa , Sandalias Fifi19 Slide de Donald J Pliner para mujer Negro Sandalias planas Softey para mujer Steve Madden Rosa / Multi Ringor Bandit 2.0 Spike- Pitching Rojo blanco Giesswein Unisex Acadia Gris Botín de hebilla de mujer Brinley Co Negro Finn Comfort Womens Sidonia Calle Patagonia, Patrón Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante marrón Zapatillas de deporte lavables Walker Propet para mujer Sr Indigo , Zapatillas de deporte Duha Fashion de Joie para mujer Oro blanco Zapatilla Deportiva Tai Hil de Tommy Hilfiger Armada

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 NIKE Tanjun B01DX7N5TE ENG Sneaker para mujer Glacier Tanjun Blue Blue Glacier Green a395962 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

Thirteen elementary straightedge 2427865
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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.