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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Malla
Estiramiento Malla superior para Un Flexible, Ligero Siente
moldeado Ofertas de jaula en el mediopié Soporte liviano
Phylon Entresuela y suela de aire Unidad de talón para peso ligero Amortiguación
Suela Flex Grooves Give Usted flexibilidad En todas direcciones
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 de running Saucony para mujer Hurricane ISO 2 Azul / Plata / Citron Zapatillas para correr Saucony Xodus Iso 2 para mujer Berry Coral Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print , Zapatillas de running adidas Alphabounce Em W para mujer Misterio Ruby / Bahia Magenta / Gray Five Zapatillas de running Nike Kaishi para hombre Lobo gris / negro-blanco ASICS Kanmei Shoe Casual para mujer Gris medio / gris medio / carbono Brooks Women s Adrenaline GTS 17 Plata / púrpura Cactus flor / Bluebird Skechers Performance Women s Go Run 400 Obstruct Gris / azul , Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Lyl Blue / Lyl Bl / University Red / Wht Zapatillas NIKE para mujer Air Max 1 PRM Running Peltre metálico / Peltre metálico Calzado de running Pearl iZUMi para mujer Em Road H3 v2 Cerise / madreselva , Zapatillas de running ASICS Gel-Contend 4 para mujer Negro , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Pavo real negro / rosa / Living Coral , Zapatillas de running ASICS Gel-Contend 4 para mujer Blanco / Aruba / Plata , Zapatillas de running ASICS Gel-Kayano 23 para mujer Silver / Pink Glow / Parachute Purple , Calzado deportivo New Balance Vazee Urge v1 para mujer Rosa / negro Zapatillas de running NIKE Air Max Flair para hombre Cool Gray / Solar Red-Black adidas Women s Edge Lux 2 Sol / Plata / Naranja Zapatillas de running New Balance 990V3 para mujer Armada Supra Martine Run Skate Shoe para hombre Melocotón tropical / carbón de leña , Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Blanco negro , NIKE Roshe G Spikeless Golf Shoes 2018 Mujeres Blanco negro Brooks Womens Asteria Plum Caspia / Diva Pink / Orange Pop Zapatillas para correr Pearl NZUMi Women s W EM Tri N 1 Blanco / Paradise Pink Under Armour Threadborne Fortis para mujer Negro (001) / Stealth Gray , Zapatillas de running adidas Women s Edge Lux W Black / Vapor Gray Metallic / Orchid Tint Zapatillas de running NIKE Air Max 2017 para mujer Blanco negro , Zapatillas de correr AS-Lite Nimbus 18 Lite-Show para mujer ASICS Negro / Plateado / Tiburón Zapatillas para correr Mizuno Women s Wave Rider 21 Trébol / blanco ,
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 Sneaky Sport de mujer de Charles David Negro Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Fiji / Coral vivo , Zapatillas náuticas para mujer Saucony Originals Púrpura , Earth Origins Sophie Women s Sandal Negro Sandalia de cuña Jubilee Propet para mujer Corteza múltiple Zapatillas de entrenamiento NIKE para mujer Duel Racer Running Pale Gray / Dust-light Bone Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Negro / Naranja , MT-onemix Lovers Calzado deportivo ligero de malla Running Jogging Sneakers Rojo blanco Mizuno Women s Wave Daichi 2 Trail Runner Remolacha / Plata adidas Performance Ultraboost X para mujer Carbono / Cristal Blanco / Núcleo Negro , Rookite 325 Trail Runner de mujeres Inov-8 Gris oscuro , Zapato para caminar WW411v2 New Balance para mujer Blanco / rosa Kenneth Cole REACTION Womens Cake Shop 2 Soft Gold Synthetic , Zapatillas de running adidas Women s Edge Rc W Gris Dos / Plata Metálica / Energy Aqua Sanda Verra Teva para mujer Surf Negro / Multi , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Armada Zapatillas de skate auténticas Vans Kids (Excedente) Vestido Blues / Grafito azul Zapato de skate Diadora B.Elite Suede Azul ara Sandalia mujer Tania Slide Plata / Metálico , Sandalia de la diapositiva de la cruz de Anica Criss Cross de las mujeres de Cole Haan Gris oscuro , Plataforma CLARKS Women s Parram Waltz Cuero blanco Sandalia Rockdale Lanea Woven Slide para mujer Latte tejido La Pamela de La Plume para mujer Beige Python , Sandalia Chloe Slide de KENDALL + KYLIE Negro , Zapato de baile femenino moderno de la gema del soleil AC de PUMA Magenta Púrpura / Magenta , Zapatilla Elodie de UGG Australia para mujer Negro Mocasín Sabrina Moc II Lamo para mujer castaña , Sandalias de vestir Patti de Sam Edelman para mujer Cuero desnudo clásico Bota corta con botones Jackie de mujer FRYE Negro - Cuero antiguo pull up lavado-76593 , CLARKS Padmora Oxford para mujeres 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.


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 Essential Men s Air Presto Essential Blustery B01AYB5HYS Presto/ Terra Orange 93ab5b5 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.