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

Saucony Kineta para Relay Runshield Running Running Shoe B01NBOMNQ7 para mujer Negro/ Menta a94969a

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,

Brooks Womens Hyperion Peacoat / Purple Wine / Aruba Blue Saucony Women s Guide 10 Zapatillas para correr Plata / Berry , New Balance Women s w990v4 Running Shoe Gray / Castlerock , Amortiguación 635v2 New Balance para mujer Plata Zapato deportivo ASICS Gel-Cumulus 18 para mujer Acuario / Flash Coral / Blue Jewel , Zapatillas de running NIKE Air Max Torch 4 para mujer Blanco Zapatillas para correr Saucony Women s Excursion Tr11 Gris rosado , Zapatillas de running adidas Ultraboost W para mujer Pizarra negra / negra / oscura Zapato de running ASICS GT-2000 4 para mujer Carbón de leña / Flash Coral / Flash Yellow , Zapato de running Saucony Zealot ISO 2 para mujer Azul marino / azul / sil Zapatillas de running Saucony Women s Cohesion TR11 Gris / morado , Zapatilla de running Pearl Road N2 v2 para mujer Pearl iZUMi Deep Lake / Lime Punch , Calzado de running New Balance para mujer 530v2 de New Balance Reflejo / Luz Porcelana Azul , Zapatillas de running Mizuno Wave Rider 20 para mujer Negro / Blanco / Rosa Glo , Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Wolf Gray / Metallic Rose Gold , Zapatillas de running NIKE Free Run para mujer - SU16-10 - Blanco Mango / Sunset Glow Zapatillas de running Reebok Women s Run Supreme 2.0 Mystic Maroon / Poison Pink / White , Zapato de running Fuzex Rush para mujer de ASICS Medio gris / Bay / Flash Coral , NIKE Men s Flex 2017 RN Negro / Hematita Metálica / Antracita / Gris Oscuro Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Silver Mink / Outerspace Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Negro / Negro-corredor Rosa-blanco , Calzado deportivo New Balance para mujer Vazee Rush V3 Blanco / Vintage Indigo Zapato de running ASICS GT-2000 4 para mujer Plata / turquesa / sacador de lima , NIKE Mujeres Air Max Thea Zapato deportivo ASICS Gel-Excite 4 para mujer Negro / Blanco / Rosa Knockout Under Armour Threadborne Fortis para mujer Negro (001) / Stealth Gray Brooks Womens Hyperion Peacoat / azul marino / cacatúa Zapato de running Fuzex Rush para mujer de ASICS Medio gris / Bay / Flash Coral , 361 Zapatillas de running para mujer 361-Enjector Negro / Blanco / 0900
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 Sandalias Terrell Ivy Lattice Sport de Merrell para mujer Sodalita , Zapatillas de running NIKE RN Motion FK 2017 para mujer gratuitas Negro / Blanco-oscuro Gris-voltio Sandalias OLKAKAI Ohana - Hombre Negro , NIKE Kids gratis Rn (grande) Plata negra / metálica Zapatillas de trail running Salomon Women s Sense Pro Trail Pulga Negra , Earth Origins Tova Azul cobalto Zapato de running Fuzex Rush para mujer de ASICS Medio gris / Bay / Flash Coral , Salvatore Ferragamo Women s Groove Bianco / Gancio Oro Spazzolato Zapatillas para caminar Merrell Siren Sport Gore-TEX para mujer - SS17 Gris Drew Womens Ariana Dusty Gold Zapatillas de golf ECOM Biom Hybrid 3 Gore-Tex para mujer Blanco / Teaberry , Sandalias Emmerly Slide de Vince Camuto para mujer Negro Sandalia para mujer FRYE Ally Criss Cross Slide Plata Sandalia de cuero Lucili de Nine West para mujer Negro / Cuero Multi , Sandalia Dolce Vita Hazle Slide para mujer Fuera de cuero blanco , Sandalia plana Dionne de Easy Street para mujer Blanco Zapato de voleibol Gel-Tactic 2 para mujer de ASICS Negro / Asics Azul / Plata 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) 6030-gold Stardust & Gold Leather Bomba de vestido Daisie de mujer Steve Madden Gris claro gamuza , Zapatillas de ciclismo de carretera Fizik R5 UOMO BOA Negro / gris oscuro Zapato de campo traviesa Kilkenny XC5 para mujer Saucony Berry | , Botas de piel de oveja real EMU Australia Womens Bells Beach Lo Winter Chocolate MONTAÑA BLANCA Cumbre de la Mujer Fantasia Boot Cuero negro Estilo suave de Hush Puppies Glynis Ii Botín para mujer Marrón oscuro Vitello / Python PU , Zapatillas de running NIKE Men s Train Speed ​​4 Negro / Garganta Verde-Barroco Marrón-bambú , FRYE Mujeres Erica Stud Oxford Negro Belleville Botas de Combate y Vuelo Impermeables, 790 Broncearse , El botín con cordones Sheena para mujer Fix y botín de moda con adorno de estrella con tachuelas Gamuza borbónica , Calzado para caminar CLARKS Women s Wave Andes Sage Nubuck , Trotters Jake para mujer plana Marron oscuro

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 Saucony Kineta para Relay Runshield Running Running Shoe B01NBOMNQ7 para mujer Negro/ Menta a94969a 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.