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.

CORRAL bordado de cuadrada mujer con de tachuelas bordado B01N8WY2S1 floral vaquera punta cuadrada - R1373 marrón 066c874

WESTERN-INSPIRED ESTILO - Cuidadosamente diseñado con intrincado bordado floral y acentos tachonados para sacar eso espíritu de vaquera en ti!
CUADRADO PERFIL DE PUNTA - Construido con una punta cuadrada perfil que agrega al oeste de los zapatos tocar y dar tus pies más espacio para respirar.
ROPER TALONES - Los zapatos se combinan con Roper de 1.5 pulgadas tacones y goma talón agarra para agregar un poco de jazz en tu ¡Avances!
PREMIUM BOTAS DE CALIDAD - Las botas de Corral compañía ha sido considerado a ser uno de los primeros fabricantes de botas en el mundo. su dedicación y alta consideración para la moda occidental, puedes estar seguro que cada par está hecho con excelente materiales de calidad y servicio.
calzado cowgirl cowboy mujer mujer mujer damas femeninas al aire libre país de recreación montando al sur caballo a caballo clásico resistente autentico genuino real hecho a mano moda de cuero deportes de viaje Tejas modernas de lujo Todos los días adulto casual mexicano botas botines trabajo estilo occidental cómodo talón zapatos patente tacones hechos a mano prima rústica vestido resistente altura maxima
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

CORRAL bordado de cuadrada mujer con de tachuelas bordado B01N8WY2S1 floral vaquera punta cuadrada - R1373 marrón 066c874 - 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

CORRAL bordado de cuadrada mujer con de tachuelas bordado B01N8WY2S1 floral vaquera punta cuadrada - R1373 marrón 066c874

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.

CORRAL bordado de cuadrada mujer con de tachuelas bordado B01N8WY2S1 floral vaquera punta cuadrada - R1373 marrón 066c874

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

Botas de vestir TuffyRider Belmont Ladies Negro 105 LD - , Adtec 9 \ castaña Botas de trabajo de hojalata South Women by SW Work Boot Multi , Blondo Women s Velma Waterproof Riding Boot Ante negro BEARPAW Women s Bethany Chocolate, caqui Bota de moda Emma Tall BEARPAW para mujer Esmeralda Bota Ecuestre Subtlewc LifeStride para mujer Marron oscuro , Noble Outfitters Mujeres en todo alrededor de las botas de flores Ladrar , Ariat Womens V Sport Tall Zip Tall Riding Negro Bota de montar de pantorrilla ancha Christine de Franco Sarto para mujer Negro , Bota occidental Roper Crossrider para mujer marrón Bota de paddock de la Fundación Dublin Childs Foundation Negro , Bota Ecuestre Subtlewc LifeStride para mujer Marron oscuro , Bota Muck Boot para mujer Arctic Sport Ii Tall Snow Boot Negro / rosa Bota Ecuestre Subtlewc LifeStride para mujer Marron oscuro , Bota de paddock Ovation Women s Sport Rider II Paddock - 469395-Brn Negro , Bota de montar Xandywc para mujer LifeStride, pantorrilla ancha Negro Twisted X Boots Womens Breast Cancer Awareness Slip On Boat Shoe Bombardero / rosa Adtec 9 \ castaña , Botas de trabajo de hojalata South Women by SW Work Boot Multi , Bota de trabajo con punta de acero resistente al agua EH PO 11 BRN de John Deere para hombre marrón Sunny Equestrian Boot de Tommy Hilfiger para mujer Gris negro Botas de vestir TuffyRider Belmont Ladies Negro 105 LD - Ariat Womens Volant S Tall Zip Tall Riding Negro , Twisted X Boots Mujeres Hooey azteca patrón lienzo ocasionales zapatos Multi , Bota Ecuestre Subtlewc LifeStride para mujer Negro Ariat Womens Heritage Contour Field Zip Tall Riding Negro Bota de vaquero western Ariat Women s Heritage Roper Marrón angustiado Botas altas Timberland Womens Lakeville Negro Full-grain / Suede ,
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 de encaje Ros Hommerson para mujer Plata Top corto de mujer Saucony Bullet Grapp Crush Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Berenjena / Sal marina Zapato de running de punto fuzeX para mujer ASICS Rosa Glo / Flash Coral / Negro , Zapatillas de deporte Onitsuka Tiger Mexico 66 Blanco azul , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar _Blanco negro , Zapatillas de deporte BZees Womens Lifetime Fabric Low Top con cremallera Multi Neon Floral , Nina Womens Contesa Suedette reflectante Platino , Zapatillas de deporte Skechers Sport Women s Empire Inside Look Gris / menta , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente de Marina Bernie Mev. Jeans Shimmer Zapatillas NIKE Womens Air Huarache Run Se Running 859429 Sneakers Shoes Hematita Metálica Negro Gris Oscuro 001 Sandalias para mujer TOMS Lexie Open Toe Casual Slingback Lona azul marino , Entrenador campeón Gusto Cross Trainer Malla rosa clara Zapato de entrenamiento WX824 para mujer New Balance Negro , Zapatillas de entrenamiento cruzado Realflex Train 4.0 de Reebok para mujer Negro / carbón / blanco Zapatillas para caminar WW669V1 New Balance para mujer Gris , Vans Unisex Old Skool Zapatillas clásicas de skate Clásico Negro / Blanco Zapatos de golf de ocio FootJoy para mujer Blanco / morado Sandalia de la diapositiva Adalea de las mujeres de Dolce Vita Gamuza naranja Sandalia Cato Dolce Vita para mujer Cuero grabado en relieve de serpiente , Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla , Así que Danca CH99 3 Heel con T-Strap Character Shoe Caramelo Dr. Martens Unisex-Adulto Pascal 8 Eye Boot Multi , Lane Sweet Paisley Mujeres Botas de media pantorrilla para mujer, Hueso Natural NIKE Kids Jr. Tiempo Rio IV (FG) Calzado de fútbol para suelo firme Negro , Drew Shoe Women s Paradise II Velco Slip-on Gris pardo , G.H. Moca MINIVOG Plataforma Wingtips Square Toe Mujer Oxfords Zapato Oro , Zapato femenino Keli de Alegria Quarry Crackle

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.

CORRAL bordado de cuadrada mujer con de tachuelas bordado B01N8WY2S1 floral vaquera punta cuadrada - R1373 marrón 066c874 - 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 CORRAL bordado de cuadrada mujer con de tachuelas bordado B01N8WY2S1 floral vaquera punta cuadrada - R1373 marrón 066c874 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 4237876
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.