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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Cuero
Cuero y suela sintética
El tacón mide aproximadamente 1 1/4
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

Zapato de para bailarina Swing Tic-Tac-Toes Zapato para mujer mujer Estaño 087a977

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,

Party Party Velvet Vestido de noche bomba Colecciones, Comodidad Zapatos de boda: Salón de baile de mujeres Zapatos de baile, medio Tacón, Tango, Suave, Estándar Terciopelo rojo 7001 Zapatillas de deporte de cuero de primera calidad para mujeres Zapatillas de deporte de neopreno elásticas con cordones de jazz Broncearse 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) 5004- Gold Sparkle , Rumpf Jive 1592 Suela delantera de gamuza Hombres negros Mujeres Jazz Hip Hop Salsa Práctica de salón de baile Luz respirable Sneaker de malla , Salón muy fino Salsa de tango latino Zapatos de baile para Mujer SERA701BBX Flate Heel + plegable Paquete de cepillo Ante negro Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 6030- Satén negro brillo , Zapatos muy finos EK8881 de la danza de salón de señoras de las señoras con 1 \ Zapatillas altas Zumba Fitness Flex Classic , Zapato de personaje de luz de posición de estudiante 650 de Capezio Negro Miller y Ben Tap Shoes; Así que Danca CH99 3 Heel con T-Strap Character Shoe Caramelo , Zapatos muy finos EK8881 de la danza de salón de señoras de las señoras con 1 \ Zapatos muy finos EK1692 de la danza de salón de baile de las señoras de las señoras con 2.5 \ Beige Marrón cuero y malla de carne , Bota de muslo Pleaser Illusion 3012 Peep-Toe de mujer Black Micro, Black Matte , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2719- Cuero plateado y malla de carne , Calzado Aria Pointe 121 para mujer Capezio Pétalo rosa Miller y Ben Tap Shoes; Salón muy fino Salsa de tango latino Zapatos de baile para Mujeres 6005 1.3 pulgadas Talón + Plegable Paquete de cepillo Satén negro , 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) 1154- Satén de marfil Zapatillas de salón de baile muy finas para mujer EK1613LED Edición limitada con 3 \ Dark Tan Satin 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) S92309 Trenza dorada y plateada , Zapatillas de salón de baile muy finas para mujer EK1613LED Edición limitada con 3 \ Dark Tan Satin Adulto 2007 \ , Party Party Velvet Vestido de noche bomba Colecciones, Comodidad Zapatos de boda: Salón de baile de mujeres Zapatos de baile, medio Tacón, Tango, Suave, Estándar 7026- Red Velvet , Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 8881- Terciopelo gris (pedido personalizado, 4-6 semanas de tiempo de proceso) 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 7013- Satén gris plateado , 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) 2824- Brillo marrón satinado y dorado Zapatillas de deporte unisex de bajo perfil Negro , Zapatillas de punta para mujer Danzcue Mango estándar duro con cinta Rosa europeo ,
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 Wstro para mujer New Balance Alpha Pink / Black / Lime Glow , Converse Chuck Taylor All Star Seasonal Color Hola LUZ ORO BLANCO NEGRO , Brooks Womens neuro 3 Laguna / Negro / Púrpura , ASICS GelGlyde Shoe para mujer Azul / plata Sandalia Yucatan ECCO Mujer Negro / topo / negro Zapatillas trail running Vasque Women s Blur SL Goblin Blue / London Fog Sandalia Merrell para mujer Terran Ari Post Sport Acuífero Skechers Men s Kirkwood One Up Negro Zapatillas de viaje Proather para mujer II Blanco , Zapatillas de deporte Arizona Trotters para mujer Negro Calzado deportivo New Balance para mujer Vazee Rush V3 Vivid Coral / Fiji , Zapatillas para correr ASICS Gel-Venture 6 para mujer Phantom / Coral / Lima Zapatillas para caminar Cinch de mujeres Devotion Plus de Ryka Gris azul , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente roja Drew Shoe Gemini cuero de mujer, malla, caucho, zapatillas de moda Black Suede / White Mesh SJP de Sarah Jessica Parker Mujer Meteor Sneaker Silver Glow Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Hueso Zapato Dantera, para hombre, L Low Waxed Court Blanco / Blanco / Negro , Sandalia de diadema Florita Carlos de Carlos Santana Blanco Zapato de Split Flex para mujeres de Bloch Dance Broncearse Sailor Moon Chibi Usa Cosplay Shoes Boots por encargo , Zapatillas de mujer Haflinger Kitty Gris claro , Salón muy fino Salsa de tango latino Zapatos de baile para Mujer S92307 2 pulgadas Talón + Plegable Paquete de cepillo Tan satinado , Bota de mujer Dansko Shirley Nubuck negro Drew Shoe Women s Fiesta Oxford Ternero negro , Mephisto Michaela Bootie para mujer Black Texas / Carla / Velcalf Premium , PIC / PAY Jonas - Cordones de las mujeres Oxford - Clásico Ropa de hombre de cuero plana Zapatos estilo holgazán (Nueva caída) Gamuza burdeos , Zapatos de tierra serval Rojo brillante , Soft Style de Hush Puppies Women s Jaylene Oxford Nubuck negro Zapato femenino Keli de Alegria Tatuaje ,

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 Zapato de para bailarina Swing Tic-Tac-Toes Zapato para mujer mujer Estaño 087a977 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 3395726
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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.