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.

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Cuero
Goma único
Ortótico soporte - Premium plantilla ortótica, incluido en el zapato, con anatómico soporte de arco y talón de amortiguación la almohadilla se ajusta a los contornos de el pie como tú caminar. comodidad y ayuda aligerar el dolor en el talón, pie, todo el camino hasta las rodillas, las caderas y parte baja de la espalda.
Fácil Gait - Ortho-Cushion ™ Sistema de peso ligero suela con amortiguación de aire suaviza paso y agrega la primavera a Tu paso. Ortho-Cushion ™ El sistema es biomecánicamente diseñado para aliviar estrés en las articulaciones, mejorar la estabilidad y ayudar a facilitar movimiento del pie cómodo y sin dolor cada paso que tomas.
No vinculante, Diseño de profundidad extra - Suave, no vinculante superior con extra diseño de profundidad y amplia puntera elimina presión sobre los juanetes y dedos de martillo. Suave, sin costuras forro interior, acolchado con espuma, elimina la presión puntos y ofertas comodidad superior y protección para pies sensibles y para aquellos que sufre de diabetes, neuropatía y artritis reumatoide.
Diseñado con innovador características de confort para el siguiente calzado: women \\ u2019s sandalias cómodas, Arco de las mujeres sandalias de apoyo, sandalias de mujer para la neuropatía, ortopedia de mujeres sandalias, mujeres sandalias ortopédicas, artritis de las mujeres sandalias, mujeres sandalias de profundidad, mujeres anchas sandalias, mujeres sandalias extra anchas, talón de las mujeres sandalias de dolor, women \\ u2019s fascitis plantar sandalias, mujeres sandalias de dolor en los pies.
Ingeniería ofrecer el mejor calzado solución para el pie Dolor, dolor en el talón Arch Dolor, Rodilla Dolor, Bola de la Dolor en el pie, metatarsiano Dolor, dolor de rodilla, Dolor de espalda y para mejorar la comodidad para pies sensibles, Pies diabéticos, Pies artríticos, Neuropatía, Plantar Fascitis, Pronación, Metatarsalgia, Neuroma de Morton, Juanetes, callos, Dedos de martillo.
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


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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

Orthofeet 12115 872 Women s Comfort B010AZMPF2 Diabetic s Extra Depth Sandal ac26290

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.


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

Wolky Comfort Jewel White Stone Print Sandalia deportiva Chaco Zvolv 2 para mujer Verdant Black , ALMA NATURAL Sandalia de pescador Beatrice para mujer Chocolate Sandalias Newport H2 de KEEN para mujer Cáñamo / Gorra blanca , Sandalias KEEN Mujer Whisper-w Higo de verano / Crabapple , Sandalia Terrell Ivy Wrap Sport Merrell para mujer Sodalita Sandalia Drew Element para mujer Nubuck azul marino , Chaco Mujer ZX1 Classic Sport Sandalia Estallido de Berry Sandalias KEEN Mujer Ana Cortez Flip-W Negro , Sandalia NIKE Benassi JDI para mujer Blanco negro , Sandalia deportiva Nolita para mujer Anne Klein Tela de peltre Sandalia atlética Columbia Slide Caprizee de cuero para mujer Ciudad Gris / Negro , Chaco Mujer ZX1 Classic Sport Sandalia Pep Pine , CLARKS Womens Wave Grip Negro sintético Sandalias KEEN Mujer Whisper-w Higo de verano / Crabapple Sandalia original Teva Women s Original Miramar Fade Dark Purple / Multi , Sandalia Merrell Tetrex Crest Wrap Sport para mujer Legion Blue Sandalia Helen de Propet para mujer Gris pardo Revs Seva Sandals, Sandalias de reflexología Para hombres y mujeres. Absorción de impactos, Cojín Comfort & Soporte del arco marrón , Sandalia Savannah de Ryka para mujer Azul Medieval / Azul Suave , Sandalia de mujer KEEN Rose Blue Opal / Provincial Blue , Sandalia deportiva Nolita para mujer Anne Klein Tela azul marino Wolky Comfort Jewel 12500 nubuckleather rojo , Chaco Z / Canyon 2 Negro acolchado , Amalfi by Rangoni Sandalia deportiva Cestello para mujer Perla / Taupe Parmasoft , Sandalia deportiva Tirra para mujer Teva Gris oscuro , MAVIRS Brand Women Sandalias 2018 Verano Correa de tobillo de flecos Stilettos 12CM Extreme High Heels Zapatos azules negros Tamaño EE. UU. 5-15 Desnudo , Sandalia Savannah de Ryka para mujer Azul marino / verde Sandalia Merrell para mujer Terran Ari Post Sport Baja Azul
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 NIKE Women s Tanjun Lobo gris / blanco , Chaco Women s Z2 Classic Athletic Sandal Edredón dorado , Sandalia deportiva Tirra para mujer Teva Teal / Dark Shadow , Sandalias Drew de mujer de Drew Cuero liso de color salmón , Zapatillas náuticas para mujer Saucony Originals Tan Cream adidas Running Womens Supernova Sequence 9 Gris sólido / plata mate / blanco , Chaco Mujer ZX1 Classic Sport Sandalia Origami Black Zapatillas de mujer Salomon X Ultra 3 W para excursionismo Sombra , BZees Sandalia con cuña de beso para mujer Negro , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Azul / blanco , Zapatillas de viaje Proather para mujer II rojo Crocs Unisex Modi Sport Slide Azul marino / gris claro Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Borgoña / ondulado / estiramiento , Hush Puppies Men s Gus Oxford Cuero negro , Easy Spirit Twist2 Sneaker para mujer Negro NAOT Kayla Elegant Sandalias de mujer Charol negro , Sandalia PORT813FLY de mujer FLY London Lápiz labial mousse rojo Zapatilla Ascot de hombre UGG Carbón , Teva Hydro-Life Ballet de mujeres Teva Negro Vans Auténtico Blue Ashes / Parisian , Supra Vaider LC Sneaker Rojo blanco Clásicos de base Slip-on (Tm) para hombres de Vans True White , Zapatillas de golf Sunnylite V2 para mujer de PUMA Blanco / Rosa Rojo / Violeta Gris , Sandalia para mujer Cait Donald J Pliner Negro Sandalia Heather Slide de mujer Vía Spiga Ladrillo de gamuza Sandalias con diseño de palmas florales Tommy Bahama para mujer Desnudo , Orthaheel by Vionic Indulge Gemma Mujeres punta redonda lona gris Slipper negro , Trotters para mujer Dea Slingback Pump Armada , Zapatillas de ciclismo Smartodoors con suela de carbono o suela de tpu de nylon para carretera y MTB Azul para MTB Propet Women s Tour Walker Strap Sneaker Blanco deporte

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.

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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 Orthofeet 12115 872 Women s Comfort B010AZMPF2 Diabetic s Extra Depth Sandal ac26290 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.

Sandalias y toboganes deportivos
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 372027
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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.