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

J. B00YBED8YG Rene233; Rene233; J. rojo ce3d578Updated 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.

J. B00YBED8YG Rene233; Rene233; J. rojo ce3d578

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

MUK LUKS Suede para mujer Amira Bootie Slipper Marron oscuro , Zapatillas Bellamo de Ted Baker para mujer Gris , Michael Kors Womens Destin Cuero tejido de rattan / yute Zapatillas Ivana de Spring Step para mujer Burdeos Vionic Women s Tide Ii Bronce Metálico Naturalizer Michelle s Dress Dress Pump Gris pardo TOMS Women Classic Slip on Lona azul marino , Orthaheel by Vionic Indulge Gemma Mujeres punta redonda lona gris Slipper Dark Blue Zebra , UGG Mujer Ansley mocasín Pizarra Drew Women s Summer Armada , Zapatillas de casa de gamuza con forro de piel sintética de mujer de HomeIdeas, mocasines de exterior de interior transpirables Vino Bomba de vestido LifeStride Sevyn para mujer Black Multi Zapatillas Ida Vionic para mujer Tela escocesa gris TOMS - Slip-On de lienzo clásico para hombre Caoba roja Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos melocotón Zapato clásico Haflinger para mujer Carbón de leña Polen Tempur-Pedic Kaydi Closed-Back Clog para mujer Negro grasiento Zapatillas Iris Easy Spirit para mujer Negro , Tobogán Chloe de Patricia Green para mujer Blanco Zapatilla de UGG Women s Lane Negro , Bomba de vestido Kalila Total Motion Kalport para mujer Barley Multi Sandalias de plataforma C-Janie de mujer de Ellie Shoes Plata , Bomba Francesca de mujer Adrianna Papell rojo Zapatilla de deslizamiento Loki para mujer BEARPAW Nuez dura Zapatillas Zeppa en suela francesa para mujer Lámina gris , Bomba antideslizante a medida Nolan para mujer Franco Sarto Negro , Zapatilla de cuña para mujer Jacques Levine # 4640 Hueso Mujeres de Modemoven Charol Bombas, Point Toe Tacones, Slingback Bombas, zapatos de noche, lindo Stilettos 10CM Azul pálido con suelas rojas Zapatilla de mujer Old Friend Scuff Púrpura

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 Zapatillas de running Adidas Performance Ultra s Boost M para hombre Blanco / gris claro / negro KEEN Sandalias con correa de mujer Maya Gris neutro / vapor , Zapatillas Nike Free TR Fit 3 para mujer 555158-402 Ale Brown / Ale Brown / Blk / Nbl Rd , SAS Womens Sunny Pecan Brown , Zapatillas de running ASICS Gel-Contend 4 para mujer Plata / aguamarina / verde fuerte Prop & eacute; t Zapatillas para caminar TravelFit para mujer Lima / Azul Sandalia universal Teva para mujer Magenta Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Blanco / Negro / Sigilo Naturalizer Samantha Pointed Toe Flat para mujer Satén negro , Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Plata / Lima Zapatillas de tenis NIKE Womens Zoom Vapor X Blue Recall / Racer Pink-white Zapatillas de yoga Vibram para mujer Alitza Loop Fitness Magenta , Vans Auténtico Blue Depths / Very Berry Onitsuka Tiger Asics Unisex Lawnship 2.0 Azul oscuro / azul oscuro , Dr. Zen Jordan Comfort Profundidad extra terapéutica para mujer Cuero para calzado / malla con cordones Negro / Plata , Sandalia Sloan Slide de Naturalizer para mujer Negro , Zapatos muy finos EK8881 de la danza de salón de señoras de las señoras con 1 \ 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 1620s- Cuero plateado , Bomba de vestimenta Cambredge para mujer Jessica Simpson Desnudo , Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 1682- Pu Beige Claro Predicciones Comfort Plus Women s Karmen Pump Patente desnuda , CLARKS Mujer Caddell Hop Boot marrón , Zapatillas de deporte Apollo Chukka de hombre nativas Regatta Blue / Shell White / Shell White Rubber , Dr. Martens unisex-adulto 1461 3 ojo zapato Nappa negro , Botas Eric Michael Laguna para mujer rojo TOMS Women Laurel Burnished Suede Boot Cuero crudo , Tobillera BT Garin de BareTraps para mujer Gris pardo , Zapatillas de deporte Easy Spirit AP1 Sport Black-microfiber Panorama de mujeres de Eastland Broncearse , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Saffron Suede

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 J. B00YBED8YG Rene233; Rene233; J. rojo ce3d578 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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