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


Zapatillas/ de running NIKE de Huarache Run SE Run para mujer Blanco/ negro-blanco/ negro 414536c - movie-store.online

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

Zapatillas/ de running NIKE de Huarache Run SE Run para mujer Blanco/ negro-blanco/ negro 414536c

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.


Zapatillas/ de running NIKE de Huarache Run SE Run para mujer Blanco/ negro-blanco/ negro 414536c

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 para mujer Lunarepic Flyknit Shield Negro / Metálico Plata-oscuro Gris-sigiloso , Zapatillas de running Under Armour Speedform Intake 2 para mujer Ivory (102) / Ghost Grey Saucony Women s Guide 10 Zapatillas para correr Azul claro / azul , Zapatillas de running New Balance 1340v3 para mujer Gris oscuro Zapatillas de running Saucony para mujer Stretch N Go Breeze borgoña Zapatillas de running Under Armour Speedform Slingshot para mujer - SS16 Puerto / Aqua Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Niebla oceánica / Blanco-azul Glow-hyper Jade , Zapatillas de running Saucony para mujer Echelon 6 Gris rosado Zapatillas de running NIKE Women s Air Zoom Structure 20 Racer Pink / White-pure Platinum Amortiguación 635v2 New Balance para mujer Plata , Zapatillas de running adidas Aerobounce Pr W para mujer Aero Blue / Aero Blue / Noble Indigo Zapato deportivo ASICS Gel-Cumulus 18 para mujer Negro / Plata / Negro , Zapatillas para correr de carretera Altra One V3 para mujer Púrpura / Naranja , Zapatillas de running W1540V2 New Balance para mujer Blanco azul Zapatillas de running NIKE Huarache Run SE para mujer Negro , Zapatillas de running adidas Originals Women s Edge Lux W Alta resolución Azul / Aero Azul / Blanco , Zapatillas de running ASICS Gel-Flux 4 para mujer Negro / Plata / Carbono , Zapatilla de running Pearl Road N2 v2 para mujer Pearl iZUMi Deep Lake / Lime Punch Zapatillas de calle YKTM Performance Women Boost 4.0 Street Running Shoe Road gris , Zapatilla de running AS-Gel-Kayano 24 Lite-Show para mujer ASICS Índigo Azul / Negro / Reflectante , Brooks Womens Hyperion Imperial Purple / Blue Jewel / Vida Nocturna , Zapatillas de running New Balance 1340v3 para mujer Blanco / morado , Zapatillas de deporte NIKE Flex Training 6 Print para hombre Zapatillas de deporte atléticas Anthrct / Pr Pltnm-sprt Gry-ttl Zapatillas de running AMPLICA para mujer ASICS Gris Medio / Plata / Aruba Azul Zapatillas de running adidas Alphabounce Rc W para mujer Indigo crudo / Gris Uno / Noble Indigo , adidas Performance Ultraboost X para mujer Misterio Ruby / Core Black / Trace Pink , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Plata negra / metálica Zapatillas de running Nike \ Volt / Black / White Zapatillas de running ST-2 para mujer Topo Athletic Frambuesa negro
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 de mujer Drew Alana Pewter-bronce , New Balance Women s w990v4 Running Shoe Negro / Plata Zapatillas de running Ultra Kalani 3.0 de Zoot para mujer Negro / Rosa Brillo / Azul atómico , KEEN Hombres Newport H2 Sandal Negro , Sandalias KEEN Womenport Newport Hydro-W Imán / Navega por la Web , NIKE Kids gratis Rn (grande) Azul profundo / blanco oscuro Zapato de running ASICS GT-1000 6 para mujer Negro / Negro / Plata , Brooks Womens Asteria Plum Caspia / Diva Pink / Orange Pop Sandalia Jacardia Merrell para mujer Armada , Zapatillas para correr Saucony Women s Cohesion 10 Armada Under Armour Women s Horizon KTV Antracita (100) / gris oscuro , Zapatillas de running definitivas Ryka para mujer Frost Grey / English Rose / Chrome Silver , Zapatillas de running Nalu para mujer Ryka Azul / plata , Adidas Originals para mujer EQT Support Adv PK W Energy Aqua / Energy Aqua / White Zapatillas de running Under Armour Speedform Intake 2 para mujer Puerto marítimo (300) / Marfil Zapatillas de running ASICS GT-2000 6 para mujer Indigo Blue / Indigo Blue / Smoke Blue , Zapatillas de running NIKE Air Max Flair para hombre Blanco Negro 101 Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Estaño J Slides Sandalia Ritsy para mujer Camello Baabuk Wool Sneaker - Mujer Gris claro / naranja , Zapatillas para caminar Ryka Women s Dash 2 Azul marino / gris / morado Converse Chuck Taylor All Star Core Ox Salsa roja / blanca Botas de tobillo Pallaphoenix K Palladium para mujer Negro Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Negro , Circus de Sam Edelman Sandalia de mujer Raquel Slide Plata blanda , Sandalias Peep Toe de Misu para mujer Latin Salsa Blackgo Practicar zapatos de baile para salón con tacón de 2.75 \ Negro , Zapato clásico Haflinger para mujer Gris Nueve West Women s Lynneah Leather Mule Luz dorada metalizada , Tobillera para mujer Easy Street Questa marrón TOMS - Slip-On de lienzo clásico para hombre Caoba roja ,

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 Zapatillas/ de running NIKE de Huarache Run SE Run para mujer Blanco/ negro-blanco/ negro 414536c 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.