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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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

Zapatos de de golf de/ ocio B01MYR3NG0 FootJoy para mujer Blanco/ morado 6b549cd

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.


Zapatos de de golf de/ ocio B01MYR3NG0 FootJoy para mujer Blanco/ morado 6b549cd

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 golf ECOM Biom Hybrid 3 Gore-Tex para mujer Coral con especias / Coral con especias Zapatos de golf de ocio FootJoy para mujer Carbón FootJoy EmPower BOA Zapatos de golf CERRAR Mujeres Plata / Lila, Plata , Zapatillas de golf Skechers Women s Go Golf Birdie Natural / Coral Zapatillas de deporte Adidas Adipure Sport para mujer Blanco / amarillo / gris FootJoy New Womens FJ emBody zapatos de golf de liquidación - Elija el tamaño! Blanco Zapatillas de golf Skechers Women s Go Golf Birdie Natural / Coral Zapatillas de golf adidas Climacool Knit Cblack / D, mujer, de adidas Luz Onix Zapatos de golf de ocio FootJoy para mujer Carbón , FootJoy Emerge Golf Shoes Liquidación 2017 Mujeres Blanco-m Zapatillas de golf adidas Wombin S Boost para mujer adidas Correr Blanco / Plata Metálica / Flash Rojo , ECCO Women s Street Evo One Luxe-w Blanco / Mineral , Zapatillas de golf Skechers Women s Go Golf Birdie Malla negra , Zapatillas de golf PUMA Golf PG Blaze Disc, para mujer Puma White-puma Silver Zapatillas de golf híbridas casuales para mujer ECCO Acuático Zapatillas de golf adidas Bounce para mujer, adidas, para mujer Núcleo Negro / Ftwr Blanco / Oro Real S Zapatillas de golf clásicas híbridas ECCO Women & rsquo; s Blanco / Teaberry , Zapatillas de golf adidas W Adipure DC de adidas para mujer Ftwr Blanco / Plata Met./ Plata Met. , Zapatillas de golf adidas Adizero Sport II para mujer Glow Purple / Runnig White / Metallic Silver Zapatillas de golf Lexington Marc Joseph New York para mujer Azulejo de arena , Zapatillas de golf Lexington Marc Joseph New York para mujer Negro Granoso Zapatillas de golf Lexington Marc Joseph New York para mujer Negro Granoso Zapatillas de golf adidas W Climacool II para mujer Raspberry Rose-tmag / Ftwr Blanco / Plateado Metálico , Zapatos de golf de ocio FootJoy para mujer Blanco / púrpura-m , Zapatillas de deporte Biom Hybrid Sport ECCO para mujer Blanco / Lime Punch Zapatillas de golf Adidas Golf Adidas, para mujer, de adidas Golf Mujer Gris / rojo noche Sandalias de golf Sandbaggers Tango para mujer Negro Zapatillas de deporte adidas Climacool Ballerina para mujer Solar Azul / Blanco / Plata Metálica Zapatillas de golf NIKE Explorer 2 de mujer AA1846-101 Blanco / Blanco / Gris Lobo / Plata Metálica ,
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 Drew Element para mujer Nubuck azul marino Sandalia para mujer Terrell Ari Wrap de Merrell Acuífero , NIKE Women s Tanjun Paramount Blue White Lava Glow , Zapatillas deportivas para hombre NIKE Lunar Apparent Cool Grey / Deep Royal Blue-pure Platinum , Supra Martine Run Skate Shoe para hombre Carbón blanco NIKE Women s Tanjun Marrón nocturno , Zapatillas para correr Mizuno Women s Wave Rider 21 Patriot Blue / White , Zapatillas de trail running Topo Athletic MT2 - Indigo / fucsia para mujer 6 - Zapatillas de running NIKE Air Max 90 para mujer Vela / radiante esmeralda negro Zapato de running ASICS GT-1000 6 para mujer Negro / Cantel / Carbono Sandalia de mujer de nivel medio Aerosoles Tela azul oscuro , Sandalias con cuña de tacón medio Lynette de Onex para mujer Blanco OLUKAI Womens Eheu Thong Sandal Negro Negro , Zapatillas para caminar ProFit TravelFit Slide para mujer Negro rojo Zapatillas de paseo Skechers Go Walk Lite-15433 para mujer Blanco negro Sandalia de tacón de cuero de maná para mujer de Nine West Cuero natural claro Zapatillas deportivas para caminar Spira CloudWalker para mujer con resortes Nimbus / Carbón de leña / Menta Wilson Womens Rush Pro 2.0 Gris / Neptuno Clásicos de base Slip-on (Tm) para hombres de Vans (Mono Canvas) Asfalto , Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre Negro Zapatillas DC Mujeres Chelsea TX SE Sneaker Estampado azul / blanco , Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre (Cheque de dos tonos) Azul imperial / Blanco verdadero , Sandalia de la diapositiva de Benicia de las mujeres de Dolce Vita Tela natural / negra , Sandalia Harlotte Woven Slide de Via Spiga para mujer Cuero multicolor , Zapatillas de triatlón Tri X-Speed ​​3 para mujer Louis Garneau Rosa resplandor / amarillo brillante Bota Deia para mujer de TOMS Desert Taupe Suede Radial Perf / Farren , Zapatos de tacón alto con tachuelas y tachuelas Kaitlyn Pan en punta Patente negra / correas negras / tachuelas de pistola de metal Zapatos con cordones Wolky Comfort Gobly Ante negro de Malibú Dr. Martens Gunby Steel Toe Shoe Negro CLARKS Women s CloudSteppers Sillian Paz Slip-on Loafer Gris ,

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.

Zapatos de de golf de/ ocio B01MYR3NG0 FootJoy para mujer Blanco/ morado 6b549cd - movie-store.online


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 Zapatos de de golf de/ ocio B01MYR3NG0 FootJoy para mujer Blanco/ morado 6b549cd 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.

Golf
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.