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 deportivas NIKE Lunar 18092 Apparent para Crimson hombre Antracita NIKE Crimson Blk Gray 87b9191Updated 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 deportivas NIKE Lunar 18092 Apparent para Crimson hombre Antracita NIKE Crimson Blk Gray 87b9191

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

Under Armour Men s Charged Bandit 3 Rhino Gray (100) / Cruise Blue , Brooks Womens Hyperion Imperial Purple / Blue Jewel / Vida Nocturna Nike Hombres Air Zoom Pegasus 33 Lobo gris / negro / gris oscuro / azul Zapatillas para correr NIKE Women s Air Vapormax Flyknit Pure Platinum / Plata Metálica Zapatillas de running NIKE Air Max Thea para mujer Blanco negro Zapatillas de running New Balance 990v4 para mujer Lago Azul / Lago Azul Under Armour Men s Speedform Intake Negro / Púrpura Rave / Penta Rosa , Zapatillas de running Reebok Print 2.0 para mujer Gable Gray / Fire Spark / Blanco / Pure Silver , Zapato deportivo ASICS Gel-Excite 4 para mujer Aluminio / Plata / Aqua Splash , Zapatillas para correr Saucony Koa Tr para mujer Gris azul , Zapatillas de running Reebok para mujer Zprint Her Elle MTM Gris Ceniza / Carbón / Vitamina C / Blanco , Zapatillas de running NIKE Flex Run 2015 para mujer Negro / Hyper Pink / Digital Pink / Silver , Zapatillas de running ASICS Gel-Cumulus 19 para mujer Aruba azul / azul de Aruba / azulejo turco Under Armour Women s Drift 2 Negro (001) / Blanco , Zapatillas de running Reebok Hexaffect 5.0 MTM para mujer Azul fresco / polvo de asteroide / plata metalizado / blanco / acero / aleación Zapatillas de running acolchadas 420v4 para mujer New Balance Gris azul , Zapato de running ASICS Gt-2000 5 para mujer Grey medio / blanco / rosa Zapatillas de running ASICS Gel-Nimbus 19 para mujer Menta / Plata / Azul , Camiseta adidas Swift Run W para mujer Núcleo Negro / Blanco / Núcleo Negro , Saucony Women s Ride 10 zapatillas para correr Azul , Zapatillas de running ASICS Gel-Kinsei 6 para mujer Gris claro / titanio / menta , Zapatillas de running con amortiguación W560v7 para mujer New Balance Armada Zapato de running Saucony Zealot ISO 2 para mujer Té / Algodón Zapatillas de running ASICS Gel-Nimbus 20 para mujer Blue Print / Blue Print Zapatillas para correr Zante V3 para mujer New Balance Piscis / Azul marroquí , Zapatillas de running AMPLICA para mujer ASICS Blanco / Plata / Porcelana azul Saucony Women s Guide 10 Zapatillas para correr Verde azulado / azul marino / rosa Zapatillas de running Questr Cc W de mujer adidas Aero Blue S, Aero Blue S, Colegiata Marina , Zapatillas de running adidas Performance Element para mujer Refresh W Gris medio Heather / Plata mate / Gris medio Heather Solid Gray ,

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 Drew Element para mujer Nubuck negro Zapatillas para correr Nike Zoom All Out Flyknit para mujer Gris negro Chaco Mujer ZX1 Classic Sport Sandalia Aguacate estático Zapatillas de running adidas Alphabounce Beyond W, para mujer Gris dos grises tres grises uno Under Armour Women s Speedform Gemini 2 Acero / Blanco / Plata Metálica Sandalia de mujer Drew Alana Pewter-bronce , Zapatillas de deporte CL Nylon MH Reebok para mujer Eco azul / azul impresionante / verde azulado / añil / blanco / dólar , Under Armour Men s Speedform Intake Negro Negro , Zapatillas de running NIKE Zoom All Out Low para hombre Sequoia / Palm Green / Pure Platinum / Black Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Black / White-volt , Easy Spirit Solana Oxford para mujeres Vino Nubuck Multi , SOREL Womens Ella Lace-Up Negro , Sandalia Tallis de mujer VANELi Multi Fucsia Blix Squama Print Zapatillas de mujer Reebok DMX Lite Slip Walking Blanco negro , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Limelight / Limelight / Safety Yellow , Zapatillas de running ASICS Gel-Kayano 23 para mujer Silver / Pink Glow / Parachute Purple Zapatillas para caminar Propell Mujer Travellite MJ Gris pardo , Zapatillas de paseo Skechers Go Walk Lite-15430 para mujer Natural Vans Unisex Old Skool Zapatillas clásicas de skate Vestido Blues Khaki , Zapatillas de tenis Clay Court 996 V3 para mujer New Balance Deep Ozone Blue / Ozone Blue , Zapatillas bajas con cordones All Star Canvas de Converse Chuck Taylor Carbón , Zapato de skate Evan Hi Wnt para mujer DC Granate , Zapatillas de deporte Vans ISO 1.5 para hombre / True White Running Trainers Wind Chime / True White kensie Sandalia Dominic Slide para mujer Plata ~ 50 tonos de bronce Zapatos de vestir de baile Colección, Comodidad Boda de la tarde Bombas: Salón de baile para mujeres Zapatos para América, Tango, salsa, swing, Theather Art por Fiesta Party (2.5 \ & 3 \ S2806 Cobre y rojo y blanco JOE N JOYCE Slippers Zuecos Zapatos Cuero sintético estrecho - Para hombre y mujer Negro , Zapatillas de ciclismo de carretera Shimano para hombre RP3 Negro , Bota de mujer FRYE Natalie Short Lug Engineer Carbón de leña suave engrasado , Botas Nelia de Coolway para mujer Coñac , Bota chukka de mujer Timberland Londyn 6 \ Hoja de uva ,

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 deportivas NIKE Lunar 18092 Apparent para Crimson hombre Antracita NIKE Crimson Blk Gray 87b9191 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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