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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Sintético
Goma suela
Robusto corredor neutral en malla con brillante superposiciones
GEL Sistema de amortiguación
Desmontable plantilla de espuma acomoda ortesis
Específico de sendero suela con varias superficies tracción
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,

Zapatillas de running New Balance Fresh Foam Lazr V1 Melocotón espolvoreado / Melocotón espolvoreado , Brooks Womens Revel Island Blue / Evening Blue / Lime Popsicle Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Antracita / Tarta , Zapatillas deportivas 574S Sport New Balance para hombre Azul negro Zapatillas de deporte Reebok Print Lite Rush Gr para mujer Gr - Blue Lagoon / Solid Teal / White , Zapatillas de running NIKE Unisex Flyknit Racer Negro / Blanco-blanco , Converse Chuck Taylor All Star Seasonal Color Hola Cielo azul Zapatillas de running adidas Alphabounce 2 M Maroon / Maroon / Mystery Ruby , Zapatillas de running NIKE Juvenate para mujer Lobo gris / blanco / gris lobo / blanco Zapatillas para correr A7200WX085 de Apex para mujeres Rosado , Zapatillas de deporte Onitsuka Tiger Colorado Eighty Five Blanco Blanco Zapatillas para correr Mizuno Women s Wave Daichi 2 True Blue / Micro Chip / Blue Topaz Zapatillas de running Adidas Performance Ultra s Boost M para hombre Blanco / Blanco / Rojo Zapatillas de entrenamiento de vuelo Jordan Nike para hombre Gimnasio Rojo Negro Gimnasio Rojo La Sportiva Lycan Running Shoe - Mujer Púrpura / ciruela Mujer adidas Originals Ultraboost St Negro / Plata Metálico / Carbono Calzado de running NIKE Kids Flex 2017 RN (GS) Pure Platinum / Black-wolf Gray Zapatillas de running Ryka para mujer Azul / Coral Zapatillas de running Adidas Performance Ultra s Boost M para hombre Claro Verde / Mineral / Choque Rojo Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Lino Verde 6367 , Brooks Womens Cascadia 13 Tinta / azul marino / rosa Correa de plataforma PUMA para mujer Satin En Pointe WN Sneaker Negro / Plata Converse Chuck Taylor All Star Seasonal Color Hola borgoña Zapato NIKE Roshe One femenino Concord / Ember Glow-sail Zapatillas deportivas 574S Sport New Balance para hombre Gunmetal / acero NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Blanco Blanco Zapatillas de entrenamiento para principiantes Primus Trio Everyday de Vivobarefoot para mujer Negro , Zapatillas de deporte Onitsuka Tiger Mexico 66 Blanco azul Under Armour Women s Micro G Limitless 2 Blanco (101) / Blanco ,
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 deportiva ECCO Terra 3S para mujer Arona / Papaya , Zapatillas de running Gobiv2 para mujer New Balance Poisonberry / Black , Brooks Women s Ravenna 8 Peacoat / Purple Cactus Flower / Fusion Coral Sandalia atlética de Rka Women s Glance Grey / Cool Mist Gray / English Lavender , Sandalia Uneek O2-w de mujer KEEN Blanco / oro de la cosecha , Zapato deportivo ASICS Gel-Excite 4 para mujer Gris medio / blanco / verde hielo , Zapatillas de running adidas Women s Edge Lux W Ash Pearl / Chalk White / Chalk Coral Zapatillas sin cordones Flex Merit para mujer Merrell Agua , Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Gunmetal / Thunder , Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Velcro Monterey Zapatillas de deporte Bay para hombre Caminando atlético Zapatos Blanco , Skechers Women s Walk Walk Lite-Daisy Loafer Armada , Zapato deportivo ASICS Gel-Excite 4 para mujer Azul índigo / azul / orquídea Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Azul claro , New Balance WX20v6 Cross Trainer para mujeres Amanecer / Gunmetal / Alpha Violet BareTraps Women s Dasie Platform Sandal Gray Bright Multi , Vans Unisex Old Skool Zapatillas clásicas de skate Llama Negro , Zapatillas de baloncesto NIKE Air Python PRM - Hombre, de punta redonda, de cuero Negro / Encías Marrón Claro Sandalia Dolce Vita Halle Slide para mujer Sky Blue Palm Print Birkenstock Mujer Mayari Birko-Flor Sandal Elegante encaje antiguo Sandalia para mujer Chandler de Carlos by Carlos Santana Rosa rubor , Sam Edelman Women s Slide Sandalia rojo Zapato de voleibol Gel-Rocket 8 femenino de ASICS Blanco / plata / blanco , Rieker Womens L.Slipper Brown Mineral fuerte Zapatilla Olive Scuff de Daniel Green para mujer borgoña Bomba de vestido animado de LifeStride para mujer Lux Navy Zapatilla Nohea de OLUKAI - Mujer Tabaco , Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos Peach Pale Pink Bota de campo soberano de Mountain Horse Negro , Botas de montar de homenaje de mujer de Dr. Scholl s Shoes Microfibra negra , Wolky Comfort Sneakers Kinetic 30210 Cuero Antracita

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 running 6296 ASICS GEL-Venture 5 Silver para mujer Frost Grey/ Grey/ Silver/ Magenta 42a8692 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

Thirteen elementary straightedge 2960231
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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.