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

Zapatillas B072LN1KW5 de running NEGRO-ANTRACITA 1979 NIKE Unisex Flyknit Racer NEGRO/ NEGRO-ANTRACITA 45d8b5cUpdated 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.

Zapatillas B072LN1KW5 de running NEGRO-ANTRACITA 1979 NIKE Unisex Flyknit Racer NEGRO/ NEGRO-ANTRACITA 45d8b5c

Chinese Handcuffs This has no lesson, only questions. It is just an interesting animated file, which invites student investigation. It has applications for geometry,

Mujer adidas Originals Ultraboost St Indigo crudo / tinta noble / naranja de alta resolución Zapatillas de running NIKE Unisex Flyknit Racer Negro / Blanco-blanco Converse Chuck Taylor All Star Seasonal Color Hola Beach Glass NIKE Kids gratis Rn (grande) Hyper Turquoise , Zapato Racquetball de Raqueta de Cuero NIKE para Mujer NIKE adidas Adizero Defiant Bounce para mujeres Flash rojo / blanco / escarlata Under Armour Charged CoolSwitch de mujer Azul , NIKE Women s Tanjun Blanco / Partícula Rose Zapatillas de running NIKE Juvenate para mujer Orquídea / Orquídea / Midnight Navy White Sandalia deportiva NIKE Kawa Slide para hombre Medianoche azul marino / blanco , Zapatillas Sequoia para mujer Mark Nason Angeles Lila Zapatillas NIKE Femenil Flex 2016 Rn Running Azul Brillo / Blanco / Azul Racer / Azul marino medianoche , Converse Chuck Taylor All Star Seasonal Color Hola Cáscara de berenjena , adidas Originals para mujer I-5923 Gris / Tiza Rosa / Negro Zapatillas de deporte Onitsuka Tiger Mexico 66 Blanco apagado / apagado blanco Calzado de running FuelCore Rushv3 para mujer New Balance Negro / Piscis Converse Chuck Taylor All Star Seasonal Color Hola Blanco óptico , NIKE Women s Tanjun Blanco Blanco Negro La Sportiva Lycan Running Shoe - Mujer Stone Blue / Jade Green Under Armour Speedform Slingwrap para mujer Marathon Red (600) / Glacier Grey , Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Mono Port Royale , Zapatillas náuticas para mujer Saucony Originals Maroon / Tan Salming Miles Running Shoe Mujeres Turquosie / rosa , Zapatillas deportivas G-Defy Galaxy Black Pink de Gravity Defyer para mujer Negro, rosa , Under Armour Threadborne Slingflex para mujer Flint / Rhino Gray / Lavender Ice , Zapato NIKE Roshe One femenino Negro / antracita / negro , Converse Chuck Taylor All Star Seasonal Color Hola Papel rosa NIKE Women s Tanjun Coral Stardust / Sail Sandalia deportiva NIKE Kawa Slide para hombre Medianoche azul marino / 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 Sandalias OLKAKAI Ohana - Hombre Kona / Kona Zapatillas de running Under Armour Charged Bandit 4 Team para mujer Acero (100) / Acero , Nike Hombres Air Zoom Pegasus 33 Lobo gris / negro / gris oscuro / azul , OluKai Ohana Sour Tart / Ice , Zapatos descalzos CVT-Hemp de mujer Vibram FiveFingers y paquete Premium Toesock Caqui , Sandalia Drew Angela para mujer Beige-multi , Wolky Comfort Jewel Caviar negro Zapatos descalzos CVT-Hemp de mujer Vibram FiveFingers y paquete Premium Toesock Caqui Zapatillas de running Adidas Performance Ultra s Boost M para hombre Claro / Gris / Claro / Gris / Gris Medio Zapatillas de running adidas Ultraboost W para mujer Unity Ink / Unity Ink / Dark Navy Sandalia Yucatan ECCO Mujer Polvo blanco , Trotters Mujer Simmy Mary Jane Flat Ante azul marino Zapatillas para caminar ProGrid Integrity ST2 para mujer Saucony Gris oscuro Ballet de niña de flores Aerosoles femenino Serpiente gris Klogs Footwear Astoria para mujer Perdiz Sandalia de pescador Yelena de mujer de Dr. Martens Negro Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Estaño Zapatillas de deporte Nike Womens Metcon 2 AMP Running 843972 Sneakers Shoes Azul resplandor / Negro Volt negro Calzado de mujer de fila Cress Walking Blanco / Highrise , Zapatillas de caminar de yoga minimalista para mujer de ala Vivobarefoot Guijarro , Zapato de skate Evan Hi Wnt para mujer DC Negro / Blanco / Negro , NIKE Blazer Mujer Low SE Premium Plataforma CLARKS Women s Delana Liri Pewter Metallic Sandalia Dolce Vita Delana Slide para mujer Lt Gold Leather , Bomba Lloret para mujer J.Renee Naranja / azul / verde , Rieker Mirjam 11 bomba para mujer Negro , Finn Comfort Women s Hanoi Oxford Nappa negro / Nubuck , Cole Haan Zerogrand Genevieve Perf Trainer para mujeres Surf Spray Nubuck perforado / Optic White , Zapato femenino Keli de Alegria Estaño , TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Henna / Blue Multi Stripe ,

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 Zapatillas B072LN1KW5 de running NEGRO-ANTRACITA 1979 NIKE Unisex Flyknit Racer NEGRO/ NEGRO-ANTRACITA 45d8b5c 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

For email to reach me, the word