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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Caucho
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.


NIKE Women s Tanjun Tanjun Paramount Blue White Blue s Lava Glow b45b4e4

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 Tanjun Racer estilo para mujer: 921668 para mujer 921668-004 Cool Grey / Binary Blue Zapatillas de running NIKE Roshe Two para mujer Atomic Pink / Sail-turf Orange Calzado de running NIKE Women s Air Relentless 6 Cool Gray / Black-pink Blast-white Brooks Womens PureFlow 7 Gris / negro / verde , Zapatillas de deporte con adidas Originals para mujer Extaball W Negro Converse Chuck Taylor All Star Seasonal Color Hola Negro / rojo / blanco Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Azul hielo / azul hielo / azul , Zapatillas para correr A7200WX085 de Apex para mujeres Rosado , Brooks Womens Cascadia 13 Gris / gris / rosa , Converse Chuck Taylor All Star Seasonal Color Hola Pulir Zapatillas de running NIKE Unisex Flyknit Racer Verde , Converse Chuck Taylor All Star Seasonal Color Hola Carbón , Zapatillas Drew Shoe para mujer Paige Ternero de hueso Zapatillas de trail running Olympus 3 para mujer Altra AFW1859F Gris / verde azulado Zapatillas de running Adidas Performance Ultra s Boost M para hombre Colegial azul marino / colegiado azul marino / azul oscuro Zapatillas de deporte Onitsuka Tiger Mexico 66 Gris claro / azul marino , adidas Adizero Hj Núcleo Negro, Naranja Solar, Ftwr Blanco I-RUNNER Elite Terapia atlética atlética extra Profundo Cuero / Velcro de malla Negro Zapatillas de running NIKE para hombre con versatilidad de tren gratis Deep Forest / Blk / Wht , Zapatillas para correr A7200WX085 de Apex para mujeres Rosado , AIR Jordan 1 Retro OG BG - 575441-106 Skechers Women s Go Run Forza 3 Teal Converse Chuck Taylor All Star Seasonal Color Hola Blanco óptico , Under Armour Threadborne Slingflex para mujer Grosella negra (923) / Marfil , Zapatillas deportivas G-Defy Galaxy Black Pink de Gravity Defyer para mujer Negro, rosa , NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Negro / Verde oscuro gris-palma Top corto de mujer Saucony Bullet Grapp Crush Sandalias OLKAKAI Ohana - Hombre Kona / Kona Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Blanco / Blanco / Verde ,
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 NIKE para hombre con versatilidad de tren gratis Negro / Volt / Naranja , NIKE Women s WMNS Air Max 270, Blanco / Blanco-Blanco Blanco / Blanco-blanco , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Phantom / Black Drew Womens Cascade Nubuck marrón , CLARKS Womens Wave Grip Cuero blanco 2 , Zapatillas ONEMIX Air Running para mujer Zapatillas deportivas atléticas ocasionales Rosado , Zapatillas de trail running Merrell Road para mujer Road Glove Dash 3 Horizon Blue / High Viz , Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Saúco , Mujer Swift Run W de adidas Originals, gris / plateado metalizado / blanco, 9 M US - Sandalia de vestir de cuero Gona de Nine West para mujer Dark Natural CLARKS Sandalia Arla Kaydin para mujer Tela elástica negra Chaco Mujer Fallon Sandal Arena , Zapatillas de running Under Armour UA Micro G Speed ​​Swift para mujer Negro Sandalia deportiva Marc Jacobs Women s Somewhere Crema Multi Zapatillas de running ASICS Gel-Cumulus 19 para hombre Azul Indigo / Negro / Rosa Cosmo , Sanda Verra Teva para mujer Negro. , Zapato para caminar WW411v2 New Balance para mujer Negro Zapatillas para mujer Del Rey TOMS Luz Marina , Supra Vaider LC Sneaker Woodthrush-white , Vans Old Skool Unisex Adults Low-Top Trainers Blanco negro , Kenneth Cole New York Kathy Slip Stretch Upper Sneaker para mujer Oro rosa Sandalia Trotters Ellie Wedge Slide para mujer Goldwash Sandalia Cato Dolce Vita para mujer Cuero grabado en relieve de serpiente , Zapato de seguridad deportivo Soyay RB191 Reebok Work para mujer Negro , mejor sentido Unisex Lana clásica Zapatilla para Adultos - Toda la lana natural - Ultra cómodo - Muchos tamaños y Colores Medoc rojo Bomba Parigi de LifeStride para mujer Negro Kiddy Botas Chelsea Hush Puppies Lindsi Bria para mujer Black Wp , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Pure Platinum / Black / Cool Gray / Wolf Gray , Trotters Liz Loafer para mujer Negro TOMS Clásicos de mujer 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 NIKE Women s Tanjun Tanjun Paramount Blue White Blue s Lava Glow b45b4e4 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.