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

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

NIKE Negro Women s Shoe Revolution 4 Running Shoe Negro B01N3LEFY4 Crimson Antracita Gris 757a104Updated 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:

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,

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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 Zapato de running ASICS Alpine XT para mujer Fucsia Morado / Fucsia Morado / Azul índigo , Calzado de running NIKE RN 2017 2017 para mujer Negro / Blanco / Industrial Azul / Racer Pink , Zapatillas sin cordones Flex Merit para mujer Merrell Armada , Calzado de running NIKE RN 2017 2017 para mujer Cerúleo / Blanco / Azul Espacial / Espuma de Menta Hush Puppies Power Walker Sneaker para mujer Blanco Sandalia de Bling con tachuelas Macayla UGG para mujer Latté , Zapatillas para caminar RYKA Women s Devotion Plus Blanco / plata cromo / almendra helada Brooks Womens Ghost 9 Negro / Sparkling Grape / Ceramic , Short NIKE Tempo para mujer Negro / Volt / Plata mate , Sandalia de cuña Gabor Women 84570-17 Samtchevreau azul KEEN Women s Uneek 8MM Sandal Negro Negro , CLARKS Sandalia de cuña Arla Jacory para mujer Elástico Fucsia Profundo , Zapato de malla Nohea de OLUKAI - Hombre Dark Shadow / Dark Shadow , Zapatillas de entrenamiento Reebok Crossfit Lite Lo TR para mujer China Rojo / Electro Rosa / Acero Zapatillas bajas con cordones All Star Canvas de Converse Chuck Taylor Carbón , Under Armour Threadborne Push para mujer Negro (001) / Acero Skechers Women s Define Sneaker Blanco negro , Hush Puppies Men s Gus Oxford Cuero negro Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Azul marino / blanco Zapatillas para adultos con hebilla deslizante, T9200 Blanco , NIKE Women s Free 5.0 Tr Fit 4 Print Obsdn / Ivry / Hypr Grp / Hypr Pnch Naturalizer Michelle s Dress Dress Pump Serpiente negra Zapatilla de mujer Spring Step Fluffball Gris OluKai Ohana Blanco / oscuro Java Bomba Stillson para mujer de Sam Edelman Malva en relieve de malva vino Zapatilla deslizante Nakiska para mujer SOREL Shale / Tarte , Apex Robyn Women s Oxford Negro CLARKS Mujer Caddell Hop Boot Armada Drew Shoe Women s Pepper Criss Cross cuero, caucho Casual Oxfords Negro , Bocca Loafer de mujer Franco Sarto 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.

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 Negro Women s Shoe Revolution 4 Running Shoe Negro B01N3LEFY4 Crimson Antracita Gris 757a104 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.

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

Geometry Construction Reference

Thirteen elementary straightedge
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and compass constructions are described and illustrated. The original version,
in Word format, can be downloaded and distributed.

Last update: March 16, 2016 ... Paul Kunkel

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