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

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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,

NIKE Mujer Air Max 97 Ultra 17 gris / obsidiana 917704-006 - , Calzado deportivo New Balance Vazee Urge v1 para mujer Rosa / negro Zapatillas para correr Brooks Cascadia 12 para mujer Peacoat / Flor de la pasión / Bluebird Short NIKE Tempo para mujer Franja negra / múltiple HOKA ONE ONE Zapatillas para mujer Arahi Azul Medieval / Lavanda Calzado deportivo New Balance Vazee Urge v1 para mujer Rosa / negro , Zapatillas de entrenamiento Nike Air Zoom Fearless Flyknit para mujer Melón Blanco / Brillante Zapatillas para correr NIKE Women s Air Vapormax Flyknit Gris oscuro / Reflejar órbita azul plateado , Zapatillas de running Nike Kids Air Max Zero Essential GS Olive-violet / Work Blue / Armory Navy , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Hot Punch / Black-aluminium-university Blue , Calzado de entrenamiento para correr NIKE Flyknit Lunar2 para mujer NIKE naranja , Zapato deportivo ASICS Gel-Excite 4 para mujer Gris medio / blanco / verde hielo Zapato de running ASICS GT-1000 6 para mujer Negro / Blanco / Atlantis , Nike Men s Air Presto Essential Binary Blue 400 , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Negro / Negro / Fantasma , Short NIKE Tempo para mujer Deporte Rojo / Blanco / Negro / Blanco Zapatillas de running Gel-Moya para mujer de ASICS Azul negro Nike Hombres Air Zoom Pegasus 33 Equipo Naranja / Blanco-negro-puro Platnum Zapatillas de running NIKE Air Max Flair para hombre Blanco negro Zapatillas de running ASICS Gel-Kayano 23 para mujer Glacier Grey / Bay / White , Zapatillas de running Kinvara 5 Saucony para mujer Plata / Vizi Coral / Citron Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Gunmetal / Thunder NIKE Men s Roshe Two Running Shoe Industrial Blue White 402 , Zapatillas para correr NIKE Women s Air Zoom Structure 21 Gris oscuro / lobo blanco gris Saucony Triumph ISO 3 Running Sneaker para mujer Blanco / verde azulado / Citron Zapatillas de running NIKE Women s Air Zoom Structure 20 Antracita / Lobo gris / Rosa fuego / Blanco Under Armour Women s HOVR Sonic NC Gris Nublado (101) / Blanco , Nike Men s Air Presto Essential Hueso claro / gris oscuro , Zapato de running ASICS para mujer Torrance Azul humo / Azul humo / 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 Nike Air Zoom Mariah Fk Racer para mujer Negro / Cumbre Negra Musgo del desierto blanco Zapatillas de running NIKE Air Max Thea para mujer Negro / antracita / blanco / gris lobo , Sirena de mujer Merrell Sandalia atlética Q2 Negro adidas Adizero Defiant Bounce para mujeres Flash rojo / blanco / escarlata , Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Negro , Zapatillas para correr Philadelphia Head s Headway 6 Gris / Coral Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Gris pardo / Vino de puerto-solar rojo , SoftWalk Women s Topeka Flat Armada Zapato de caminar de la correa de la andadera de la estabilidad de las mujeres de Propet Blanco Easy Street Adagio Wedge Slide para mujeres Patente / estiramiento rojo , Calzado deportivo New Balance Vazee Urge v1 para mujer Púrpura / amarillo , Zapatillas para mujer Reebok Zprint Her MTM para mujer Negro / Gris Ceniza / Blanco Zapatilla para mujer Dr. Comfort Katy Turquesa , Under Armour Women s Charged 24/7 Mid NM Acero (035) / Gris Nublado Zapato de Cross-Trainer de las mujeres de Ryka Gris oscuro adidas Athletics 24/7 Tr M Cross Trainer Core Black / Core Black / Hi-res Blue Onitsuka Tiger Asics Unisex Lawnship 2.0 Borgoña / Borgoña Zapato de skate casual Heathrow DC para hombre Azul marino / caqui Zapatillas de skate solar NIKE SB Check para hombre NIKE Negro , Zapatillas de golf Go Golf Blade de Skechers Performance para mujer Gris azulado Sam Edelman Sandalia Gio 2 Slide para mujer Negro / Multi , Sandalia Corey de Dolce Vita para mujer Rafia negra múltiple Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla Zapatillas de voleibol Mizuno Women s Wave Hurricane 2 Blanco negro , Bota de montar Xandy para mujer LifeStride Gris oscuro 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) 7016- Oro Trotters Lucy Ballet para mujer Hueso Bella Vita Womens Mel-Italy Gamuza italiana gris topo , Georgia GB00157 Oxford Marron oscuro Timberland Mujeres Eivissa Sea Oxford Lona azul marino ,

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 Adidas Women s EQT Support ADV correr Originals Blanco Zapatillas Core para correr Correr Blanco/ Core Black f5bb4d3 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

Thirteen elementary straightedge 781257
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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.


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