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.

Rieker Beige Mirjam B072K1R687 11 Rieker bomba para mujer Beige c14b305

sintético suela
Cuero suave superior
Perforado diseño con festoneado bordes
Velcro cierre
Contorneado plantilla
apilada tacón
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

Rieker Beige Mirjam B072K1R687 11 Rieker bomba para mujer Beige c14b305 - movie-store.online

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

Rieker Beige Mirjam B072K1R687 11 Rieker bomba para mujer Beige c14b305

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.

Rieker Beige Mirjam B072K1R687 11 Rieker bomba para mujer Beige c14b305

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

Cobb Hill Rockport - Aubrey Dress Pump para mujer Armada Pantuflas Aquazzura Womens Powder Puff Flat Red , TOMS Clásicos de mujer Lana texturizada negra , Vionic Womens Kari Cable Taupe , Trotters Lucy Ballet para mujer Hueso Zapatillas Zeppa en suela francesa para mujer Gunmetal Nappa Zapatillas de deporte de verano Breezy Dualyz Fit Breezy con suela extraíble Negro / verde azulado Zapatilla de zapatilla de zapatilla 100% auténtica de lujo de las señoras de Snugrugs con suela suave castaña Bomba de vestido de pasión Easy Street para mujer Blanco Trotters para mujer Dea Slingback Pump Oro Acolchado TOMS - Slip-On de lienzo clásico para hombre Universidad Negro , Pantuflas Haflinger Womens Everest Fundus Wool Pantalones , Pantuflas Matilda de Katy Perry para mujer Negro , Trotters para mujer Dea Slingback Pump Negro Negro , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Negro , Zapatilla metálica Dakota de UGG para mujer Negro Bomba de vestido Daisie de mujer Steve Madden Patente roja , Zapatilla Dakota de las mujeres de Australia UGG Regata TOMS - Slip-On de lienzo clásico para hombre Negro clásico , Zapatos planos de tacón bajo YDN para mujer con tacón bajo, deslizamiento en zapatillas Bowknot Zapatos antideslizantes de verano en deslizamiento Gris , Flip Flop FitFlop Carmel Toe para los pies de las mujeres Blanco Urbano Zapatilla Glamour de mujer verde Daniel Black Kidskin , WoolWorks Modelo 9-75B Zapatillas de piel de oveja australiana para mujer Lana de gamuza , Zapatos OLUKAI Pehuea - Mujer Stormy Blue / Stormy Blue , FitFlop Trade; Carbón Motociclistas mujeres de SKECHERS - Decisión dividida Cuero carbón / malla , Onlymaker Mujeres High Peep Peep Puntera plataforma Stiletto Tobillo cruzado Correa Hebilla Snap Vestido fiesta tacón Sandalias Multicolor Bomba de cuña Sandra Sandra Dansko Nubuck negro Zapatillas de casa de lana de invierno U-lite cálido cálido Shearling de las mujeres, zapatillas de lana caliente para las mujeres Castaño (gamuza de cuero de vaca) ,
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 Under Armour Women s Thrill 3 Grosella negra / blanco / rosa penta , Chaco Women s Z1 Classic Athletic Sandal Negro , NIKE Women s WMNS Zoom Fly, negro / blanco Vasto gris Sandalia atlética Salomon Women s Crossamphibian Swift W Mallard azul , NIKE Mujeres Air Zoom Pegasus 33 Rosado Sandalia deportiva Terrell Ari Lattice de Merrell para mujer Helecho , Sandalia deportiva Nolita para mujer Anne Klein Tela azul marino , Brooks Women s Ravenna 8 Peacoat / Purple Cactus Flower / Fusion Coral , Brooks Womens Neuro 2 Negro / Púrpura Cactus Flor / Blanco , Sandalia de 2 piezas Cobb Hill Women s Hollywood Cuero caqui Zapatillas clásicas Vionic para mujer Walker Blanco / rosa Apéndice Negro , Empuje cargado de Under Armour para mujer Bayou Blue (953) / Blue Infinity , Zapatilla Reebok para mujer ROS Workout TR 2-0 Cross-Trainer Negro / Aleación / Blanco Entrenador cruzado New Foam Crush V1 de New Balance para mujer Dusted Peach / Sunrise Glo , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Armada Zapatillas DC Beach Trase TX Skate Rosa , Zapatos de mujer de neopreno clásico Pedors Negro , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker rojo Zapatillas de baloncesto PUMA Women s Sky II HI Futur Minimal WN Cereza de Barbados Zapatillas de baloncesto NIKE Air Python PRM - Hombre, de punta redonda, de cuero Blanco , Sandalia de la diapositiva de la jalea de Leadcat de las mujeres de PUMA Puma White-puma White Sam Edelman Sandalia Bernice Slide para mujer Saddle Vaquero Saddle Leather , Stuart Weitzman Sandalia Rosita para mujer Lápiz labial de color topo , Zapatillas de voleibol adidas Crazyflight X 2 de mujer Choque amarillo / ceniza plateado / blanco , Bomba de cuña Yaralla J.Renee para mujer Taupe metálico , Minnetonka Sandalias de cuña Blaire para mujer - 71330Blk Grey Suede , Zapato clásico Haflinger para mujer Gris Eastland Solstice MLB Women s Oxford St Louis Cardinals-red , Zapatillas para caminar Slip-On Go Walk de Skechers Performance para mujer Gris

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.


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.

Rieker Beige Mirjam B072K1R687 11 Rieker bomba para mujer Beige c14b305 - movie-store.online

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 Rieker Beige Mirjam B072K1R687 11 Rieker bomba para mujer Beige c14b305 it out, and I got a chance to try out this new graphics software.


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 2004785
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.