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,

Naturalizer Samantha Pointed Toe Flat para mujer Brocado , Easy Spirit Women s Traveltime Mule Cuero blanco / azul claro , Zapato de malla Nohea de OLUKAI - Hombre Carbono / rojo oscuro , Zapato de bicicleta Five Wrest Kestrel Lace WMS para mujer Maroon / Onix Easy Spirit Women s Traveltime Mule Suede claro natural / blanco Tierra Everwood Gris ceniza , Baabuk Wool Sneaker - Mujer Gris claro / negro , Zapatilla Ascot de hombre UGG Carbón , Easy Spirit Women s Traveltime Mule gris , Zapato de malla Nohea de OLUKAI - Hombre Trench Blue / Dive Camo Naturalizer Samantha Pointed Toe Flat para mujer Cuero negro Zapatilla Ascot de hombre UGG Negro Negro Zapatilla Ascot de hombre UGG Negro , Easy Spirit Women s Traveltime Mule Cuero blanco / azul claro , Easy Spirit Women s Traveltime Mule Marfil , Easy Spirit Women s Traveltime Mule Cuero blanco / azul claro , Easy Spirit Women s Traveltime Mule rojo , Sandalia Uneek o2-w de mujer KEEN Puerto / blanco , Easy Spirit Women s Traveltime Mule Marfil , Tierra Everwood Cuero de becerro negro Naturalizer Samantha Pointed Toe Flat para mujer Gris pardo Tierra Everwood Negro , Zapato con cordones Ziane 116 1 de Lacoste para mujer Textil azul claro Easy Spirit Women s Traveltime Mule Cuero negro / negro , Zapatillas de deporte vanguardista Speeds Varsity Vanguard para mujer Brooks Denim Blue , Easy Spirit Women s Traveltime Mule Cuero negro / negro , Zapato de malla Nohea de OLUKAI - Hombre Blanco , Baabuk Wool Sneaker - Mujer Gris claro / negro Zapatilla Ascot de hombre UGG Bubbly / Sahara ,
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 Chaco Mujer Zcloud Sport Sandal Agrupado azul , Converse Chuck Taylor All Star Seasonal Color Hola Zafiro rosa Zapatillas deportivas 574S Sport New Balance para hombre Blanco New Balance WT510RS3 Trail Running Shoes para mujer Nimbus Cloud , Zapato de running Mission XR para mujer Salomon Celedon / Papaya / Pop Green , Earth Origins Tova Hibiscus Red , NIKE WMNS Zoom Fly Women 897821-411 Equator Blue / White , NIKE Women s Flex Experience RN 5 Prem Running Calzado-Negro / Rosa Negro / rosa , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Pavo real negro / rosa / Living Coral Drew Shoe La alegría de las mujeres Combo oliva , Tecnología Vionic Men s Orthaheel White / Navy Walker Equipaje bronceado Sandalia de 2 piezas Cobb Hill Women s Hollywood Cuero caqui Drew Shoe Naples Diabético terapéutico para mujeres Depth Extra Leather Shoe-On marrón , Skechers para el trabajo de la mujer Albie Walking Shoe Negro , Orthofeet 851 Comfort Diabetic Therapeutic Profth Shoes para mujer Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Rosado , Zapatillas Adidas W Tech Response Ftwwht / Ft Golf Shoe, para mujer Blanco , Sandalia de raso tejida Joanne de Kenneth Cole New York para mujer Armada Plataforma CLARKS Women s Delana Liri Pewter Metallic , Sandalia Carlos de Carlos by Carlos Santana para mujer Negro , Sandalia Evelia Slide de Bandolino para mujer Tela de peltre Lauren by Ralph Lauren Sandalia mujer Davan Slide Deep Saddle Tan Super Soft Leather , A.98 Botas de moto Isha para mujer Broncearse J. Rene233; Gris pardo Wolky Comfort Lace Up Shoes Holandés 10070 negro nubuck , TOMS Kids 10009918 Alpargata-K Rosa naranja , Zapatillas de deporte lavables Walker Propet para mujer Sr Indigo Trotters Liz Loafer para mujer rojo Slip clásico de mujer de SAS en Moca , Zapato femenino Keli de Alegria Quarry Crackle

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 Me Me Too Women Lightning s Lightning 2 Pump 19413 Nappa Metálica Rosa Dorada ef66a24 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.

Atlético
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1526741
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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.