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
Importado
Caucho único
Impacto Sistema de orientación (I.G.S) (Corriendo) : Diseño de ASICS filosofía que emplea enlaces componentes para mejorar el pie marcha natural de golpe de talón a despegar.
FluidRide: FluidRide proporciona la combinación definitiva de rebote y amortiguación propiedades con peso reducido y excepcional durabilidad.
Rearfoot y antepié GEL Sistemas de amortiguación: Atenúa el shock durante el impacto y fases de despegue, y permite el movimiento en múltiples planos como las transiciones del pie a través de la marcha ciclo.
Orientación Trusstic: Esto Sistema Trusstic: integra orientación Construcción de línea para una marcha mejorada eficiencia mientras proporcionar el mediopié integridad estructural.
sin costura Construcción
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

Zapato deportivo ASICS ASICS Gel-Cumulus Rosa 18 para mujer 16140 Diva Rosa/ Plata/ Coral 8f6cbdf

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 NIKE Air Max Thea Ultra FK para mujer Brillante Crimson / Wolf Gray-black-white Zapatillas de running adidas Alphabounce Rc W para mujer Núcleo Negro / Plata Metálico / Gris Cinco Zapatillas de entrenamiento NIKE Metcon Dsx Flyknit para mujer Negro / Cool Gray-hyper Punch , Zapatillas de running adidas Alphabounce Beyond para mujer Blanco / Gris / Gris , Zapatillas de running New Balance Urgev2 para mujer Alpha Pink / Vintage Indigo , Zapatillas para correr ASICS Gel-Venture 6 para mujer Castlerock / Silver / Honeydew PUMA Men s Roma Basic Sneaker Blanco / Alto Riesgo Rojo / Negro , Zapato de running FUZETORA para mujer ASICS Azul oscuro / azul oscuro / rosa begonia , Mujer adidas Cf Racer Tr W Gray One / Gray Two / Vapor Grey Zapatillas de running ZIGOR para mujer Zapatillas de mujer Flyknit Epic React Rosado Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Acero / Trueno , Zapatillas de running adidas Adaptive Adapt para mujer Negro / Carbono / Blanco , NIKE Women s Flex Experience RN 5 Prem Running Calzado-Negro / Rosa Negro , Zapatillas de running adidas Ultraboost W para mujer Blanco / Blanco-2 / Blanco , Zapatillas de running NIKE Air Pegasus + 29 Team para mujer Zapatillas Nike Lunarglide 9 Pure Platinum / Blanco / Blanco para mujer - Zapatillas de tenis ASICS Gel-Game 6 para mujer Blanco / Diva Rosa / Plata Zapatillas de running Saucony Women s Jazz Low Pro Gris blanco Calzado de running NIKE Kids Roshe One (GS) Lobo gris / blanco NIKE W Roshe One Womens 844994-601 Partícula rosa / vela negra , Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Phantom / Black / White Munsell Zapato deportivo ASICS Gel Quantum 360 para mujer Verde fuerte / Verde jazmín / Amarillo intermitente Zapatillas de running Brooks Women s Glycerin 16 Road Ébano / verde / negro , Zapatillas de running Mizuno Wave para mujer Inspire 14 Dapple Gray / Clover , PUMA Men s Roma Basic Sneaker Blanco-gris claro , Zapatillas de running New Balance 990V3 para mujer Negro Skechers Sport Skech Air Run High Fashion Sneaker Negro / rosa caliente , Zapatillas de running NIKE Air Max 90 para mujer Blanco negro Calzado de running New Balance para mujer Vazee 2090 Rosa / amarillo ,
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 KEEN Sandalias Newport Eco-W para mujer Negro / Imán KEEN Hombres Newport H2 Sandal Negro pizarra / Cantón Zapatillas de trail running Topo Athletic MT2 - Hombre Gris negro Sandalias Newport H2 de KEEN para mujer Plaza Taupe / Provincial Blue Zapatillas de running ASICS Gel-Kayano 23 para mujer Poseidón / Plata / Cacatúa , Skechers Cali Sandalias Beverlee Smitten Kitten Wedge para mujer Taupe Cork , Naturalizer Womens Canary Cuero negro , Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul Púrpura / Regatta Azul / Blanco Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Equipaje Tan , Calzado de caminata Allrounder by Mephisto para mujer Firelli Tinte naranja quemado lavado Sandalia Tessa de mujer Miz Mooz Moho , Drew Shoe Women s Tulip Comfort Brown Croc , Zapato para caminar WW877 para mujer New Balance Plata Zapatillas de skate casual unisex Heathrow Prestige para hombre de DC Negro / Marrón / Blanco , Zapatillas de skate solar NIKE SB Check para hombre NIKE Blanco negro Toalla de plataforma ECCO Women s Touch Plateau Platform Slide Rose Dust Sandalia de vestir Lauren de Jack Rogers para mujer Oro plateado , Sandalia Sharpie de mujer Steve Madden Color de malva Sandalia Bobbi Slide de Dolce Vita para mujer Ante azul marino , Sandalia de diapositivas para mujer Daydream Shoes de Dr. Scholl s Shoes Monument Action Leather Sam Edelman Women s Weslee Slide Sandal Ante negro , Zapatillas de volante adidas Bounce 2 de Crazyflight para mujer Blanco / Plateado Metálico / Gris Twisted X Ladies Brown / Purple Zapatos casuales marrón Bota de nieve alta Muck Boot Reign para mujer Negro Lane Junk Gypsy de Chocolate Dirt Road Dreamer Boot Snip Toe - Jg0003b Chocolate , Birkenstock womens Arizona de Sandalias de cuero XYD Casual Botines Slouchy Botas de tacón alto con punta redonda de gamuza para las mujeres Negro , TOMS Womens Desert Wedge Bootie Gris pardo , Sandalia de vestir Carrson de mujer Steve Madden Plata Bota de montar Ritten de Lucky Brand para mujer Negro

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 deportivo ASICS ASICS Gel-Cumulus Rosa 18 para mujer 16140 Diva Rosa/ Plata/ Coral 8f6cbdf 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 796118
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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.