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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Tela o Textil
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
Fibra parte superior tejida
Construir para la comodidad y transpirabilidad
Perfecto para correr y el gimnasio
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,

New Balance Women s Wlazrhe Púrpura Converse Chuck Taylor All Star Seasonal Color Hola Cáscara de berenjena Zapatillas de running Spira Scorpius II para mujer gris Zapatos Go Walk Go Step Original de Skechers para mujer Negro Negro , Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Blanco negro Zapatillas de volante Nike Women s Zoom Hyperface 2 Blanco / Juego Real Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Port Royale Zapato de running ASICS GEL-Venture 5 para mujer Frost Grey / Grey / Silver / Magenta Zapatillas náuticas para mujer Saucony Originals Tan Cream , Zapatillas de running Skechers GOrun Forza 2 para mujer Rojo / Azul (Houston 2017) Zapatillas náuticas para mujer Saucony Originals Azul marino / blanco Roxy Women s Set Session Zapatillas atléticas para caminar Antracita NIKE Women s Tanjun Marrón nocturno Zapatillas deportivas 574S Sport New Balance para hombre Driftwood / White New Balance Wl574v2 para mujeres Negro Zapatillas Drew Shoe para mujer Paige Ternero de hueso , Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Clay Green / Barely Gray 300 , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Gray Two / Gray Two / Trace Gray Carro de mujer Brooks Limón Cromo / Verde Verdadero / Negro , NIKE Mujeres Air Zoom Pegasus 33 Brillante Crimson / Gym Red / Bright Mango / White , Sperry Top-Sider Women s Crest Vibe Sneaker Armada , Zapatillas de deporte Onitsuka Tiger Mexico 66 Gris vaporoso / Peacoat Adidas Adizero Shotput Running Shoe Núcleo Negro, Ftwr Blanco, Naranja Zapatillas NIKE Flex Adapt TR Cross Trainer para mujer Gris oscuro / Blanco / Negro / Sigilo Zapatillas de running NIKE Zoom para mujer Gris Zapato deportivo ASICS Gel-Excite 4 para mujer Azul índigo / azul / orquídea , Zapatillas de running definitivas Ryka para mujer Frost Grey / English Rose / Chrome Silver Zapatillas para correr Mizuno Women s Wave Paradox 3 Capri-rosa Zapato deportivo ASICS Gel-Excite 4 para mujer Uva / Plata / Begonia Rosa ,
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 New Balance 990V3 para mujer Armada , Zapatillas de running W775V2 para mujer New Balance Azul blanco Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Negro / Vivid Pink Zapatillas de running NIKE para hombre con versatilidad de tren gratis NEGRO NEGRO , Carro de mujer Brooks Negro , ONEMIX Zapatillas para correr Air Cushiong para mujer, Zapatillas deportivas deportivas ligeras Negro rojo Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco azul Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Rough Green / Black / Palm Green Zapatillas de senderismo V753WW085 para mujer Apex gris Zapatillas de agua Merrell para mujer All Blaze Sieve Mary-Jane Azul , Dr. Comfort Victory Profundidad extra terapéutica para mujer Athletic Shoe leather / mesh cord-up Blanco , Bomba de espejos Zahn de Naturalizer para mujeres Porcelana , Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Plata blanca Zapatillas bajas con cordones All Star Canvas de Converse Chuck Taylor Negro , Zapatillas de deporte especializadas Gola para mujer Cielo azul / blanco , Zapato de entrenamiento transformado Vazee 77v1 para mujer New Balance Blossom / Graphic Vans Old Skool Unisex Adults Low-Top Trainers Gris / Violeta , Sandalia de cuero Nirveli de Nine West para mujer Cuero natural oscuro , Sam Edelman Sandalia Glenn para mujer Cuero metalizado opaco oscurecido en oro fundido Sam Edelman - Sandalia de mujer Liliana Slide Ante negro Very Fine Dance Shoes 2707 (Grado de competición) 2.5 \ 2707- Listón blanco satinado y plateado , 50 sombras de blanco 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) 1606- Puntos blancos / negros Bota ecuestre de mujer Sam Edelman Penny Cuero de whisky , Trotters Lucy Ballet para mujer Coñac , TOMS - Slip-On de lienzo clásico para hombre Universidad gris oscuro , TOMS Clásicos de mujer (Armada) Propet Cush N Foot Slipper & Oxy Cleaner Bundle Merlot Corduroy , Walking Cradles Ante Ante de las mujeres Cuero negro Hush Puppies Chardon Oxford para mujer Gris hielo , Zapatillas Slip-On Roamer SAS para Mujer Lino

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 Calzado rápido de running rápido para B01MXEJKF8 mujer flexible Reebok, tejido flexible Colegial azul marino/ Laguna azul/ Raincloud/ Blanco 34b3b05 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.

Corriendo
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.