GXWeb Polygon Playground 1

 

---

---

 

Show Start

Games 1 and 2: Dealing with Diagonals

Back to Top

Jump to Model

Challenge 1

Two teams (A and B) of $n$ players each are seated around a $2\cdot n$-gon polygonal table. Team A are allocated the odd-numbered seats, and team B, the even numbers. How do we set up the draw so that each member of team A competes against each member of team B exactly once? How many times must they meet?

---

Challenge 2 Consider a clock face with $2\cdot n$ "hours" marked around the dial. Let $a(n) =$ number of ways to match the even hours to the odd hours, modulo rotations and reflections. OR... Given a $2\cdot n$-gon... Draw $n$ diagonals so that they don't share end points, and there are an odd number of polygon sides between the end-points.

---

How many such variants do you think can be found for a hexagon ($n=3$)? For an octagon ($n=4$), decagon ($n=5$) - or the standard clock dodecagon ($n=6$)? And why stop there - try $n=7, 8, 9, 10$.

---

Below you will find $n$ draggable segments. Use these to find as many solutions (variants) for these challenges as you can. (Vary the number of sides using the $n$ controls and browse some sample solutions using $Variants$)

(NOTE: To model Challenge 2, use negative values of $v$).

To begin, you might try to:

1. drag the segments to build three different variants for the hexagon.

   Press Check to check your models.

2. drag the segments to build a few different variants for the octagon.

   Press Check to check your models.

3. drag the segments to build some different variants for the decagon.

   Press Check to check your models.

4. Try a few more, browse the samples, and then try the Quiz!

Hide Start

---

---

This document requires an HTML5-compliant browser.

( - ) (n) ( + )
$$n$$							▶
		0		6	10
$$Variants$$						▶
			-80			0	10
( - ) $Variants$ ( + )
r				▶
		0		10	10

Plot my variant ⇑Check ⇑Test yourself

RESET

---

App generated by Geometry Expressions

Back to Top

Jump to Model

 

---

---

 

Show

How many ways did you say?

Back to Top

Jump to Model

---

(Sneak Peek at the Decagon?)

---

Show Challenge 1

Challenge 1

How do we set up the draw so that each member of team A (odd numbers) competes against each member of team B (even numbers) exactly once?

For n = 1, there is only one way. For n = 2, there are 2 ways. For n = 3, there are 3 ways. For n = 4, there are 4 ways. For n = 5, there are 5 ways. For n = 6, there are 6 ways. For n = 7, there are 7 ways. For n = 8, there are 8 ways. For n = 9, there are 9 ways. For n = 10, there are 10 ways.

---

Back to Top

Jump to Model

---

Show Challenge 2

Challenge 2

List the number of ways to match the even hours to the odd hours on a $2\cdot n$-gon "clock face", modulo rotations and reflections.

For n = 1, there is only one way. For n = 2, there is 1 way. For n = 3, there are 3 ways. For n = 4, there are 5 ways. For n = 5, there are 17 ways. For n = 6, there are 53 ways. For n = 7, there are 260 ways. For n = 8, there are 1466 ways. For n = 9, there are 10915 ways. For n = 10, there appear to be 93196 ways.

---

Back to Top

Jump to Model

Hide

 

---

---

 

Show

Game 3: GXWeb Polygon Angles Theorem Generator

Back to Top

Jump to Model

Of course, polygons by their nature have TWO features which make them interesting: sides and angles. If we are going to play with polygons, it would be a shame to only play with their sides and neglect the many questions posed by their angle relationships!

For any given polygon, how many different angle theorems can be found?

Thanks to Phil Todd, creator of Geometry Expressions and GXWeb, this amazing GXWeb model automatically generates angle theorems for hexagons, octagons and more!

Explore... Try the challenges of the full GXWeb Polygon Theorem Generator!

Hide

Back to Top

Jump to Model

 

---

---

 

Show

Game 4: GXWeb Cyclic Polygon Constructions

Back to Top

Jump to Model

Finally, our Polygonal Playground would hardly be complete without some thought given to actually constructing those polygons!

Two methods are shown here, both made relatively simple using GXWeb's constraint-based geometry system and computer algebra foundation. Study and play with both - then try them yourself using GXWeb

Method 1: Quick Cyclic Polygon Construction

Back to Top

Jump to Model

---

Method 2: General Cyclic Polygon Construction

Begin with a square ABCD, and mark the center point (4). Note that the side length (AB) of the square will also be the side length of each subsequent polygon. Draw an arc of radius AB from a base vertex of the square, and mark the point (6) such that the segment (46) is perpendicular to the base. Bisect the segment 46 (point 5). Use this distance 45 to continue placing the center points of subsequent polygons above.

All the ingredients are now in place: construct a circle at each of the center points, and then step around each circle in steps of length AB!

Hide

Back to Top

Jump to Model

 

---

---

 

Show

Construct your own Model with GXWeb

Back to Top

Jump to Model

Hide

 

---

---

 

Show

Share Your Results

 

Enter your name:	? Abel, Nils Agnesi, Maria Bernoulli, Daniel Cantor, Georg Descartes, Rene Euler, Leonhard Fermat, Pierre Galois, Evariste Germain, Sophie Hypatia Johnson, Katherine Liebniz, Gottfried Lovelace, Ada Monge, Gaspard Newton, Isaac Noether, Emmy Pythagoras Ramanujan, Srinivasa Riemann, Bernhard Sylvester, James Tao, Terence Weierstrass, Karl' Zeno
My class:
Email to:

Session report length: 0 characters.

⇒ Web

⇒Email

Clear

Back to Top

Jump to Model

 

---

 

Hide QR

Share with QR Code

Back to Top

Jump to Model

QR Codes are a great way to share data and information with others, even when no Internet connection is available. Most modern devices either come equipped with QR readers in-built, or freely available. The default link here is the GXWeb Jigsaws and Quizzes, but you can use it as an alternative to sending your assessment data via email, web or share with others in your class. You can even use it to send your own messages!

GXWeb Jigsaws and Quizzes Send Your Own QR Code Reset QR Code

Send Web Address ⇑

Hide

Back to Top

Jump to Model

 

---

---

 

Show

Behind the Scenes

 

View JavaScript GXWeb Code (gxpoly.js)

 

---

 

Learn How: Create your own GXWeb Assessment Task

Learn How 2: Take GXWeb to the Next Level

Back to Top

Jump to Model

 

---

 

©2023 Compass Learning Technologies ← Saltire Software → GXWeb Polygon Playground 1

 

---