June 2024
Welcome to EFM's June Newsletter!
It is essential that every caregiver in the world reads books and does math with their young children!
EFM believes in every child’s mathematical right to equity, opportunity, and personal fulfillment.
News
Donation and Playing Cards – EFM received $5,000 from a supporter. We plan to use this money to help us give away playing cards and storybooks. Please write to me if you are interested in getting some of these materials.
EFM on Mobile Phones – The EFM mobil app is now available on iOS as well as Android phones. It includes both English and Spanish, though the Spanish material is lagging behind a little. Look for “Early Family Math” in either app store.
EFM Beginning Storybooks – These 26 stories are now available in English and 7 other languages using human translations – Spanish, French, Chinese (Simplified and Traditional), Korean, Japanese, and Arabic.
Topology for Preschoolers – Part I
I did my PhD dissertation in the area of Algebraic Topology, so Topology and Number Theory (last month’s topic) are areas of mathematics that are near and dear to me. You and your child do not need to be working on your PhDs to enjoy a lot of fun and interesting topics in Topology.
Topology is all about understanding and describing shapes, and deciding what makes two shapes the same. If you have two very different looking shapes, such as two knots, how can you tell if they are the same? What does it even mean for them to be the same?
Topology usually considers two things to be the same if you can smoothly bend, stretch, or squash one to become the other. The standard example is that a teacup is the same as a donut. If you think of the teacup as made of clay, you can smoothly deform and squash the bowl of the cup until the teacup is just a handle. At that point, you have a donut!
Topology considers spaces of any number of dimensions. In this two part series we will be looking at 1-dimensional and 2-dimensional spaces. This month we will have lots of fun with curves, and next month we’ll explore surfaces.
Simple Closed Curves
A simple closed curve is a curved line whose end is attached to its beginning, and the curve does not intersect itself. A very important property of such curves in a plane or on a sphere (the surface of a ball) is that they break the surface into three parts – the curve and the two spaces on each side of the curve. Note that this separation property does not work on a torus (the surface of a donut) – there are many simple closed curves on a torus, such as the curve that goes around the hole, that only break the torus into two parts.
Investigation – Insides and Outsides
Suppose you have a simple closed curve drawn on a piece of paper. If you pick some point, what is an easy way to tell whether it is on the inside or outside of the curve? If your curve is a circle, then it is easy enough. Suppose it is a very complicated curve, similar to the second picture. Which point is on the inside of the curve – A or B?
To answer the question, you could go winding around seeing if the point is connected to the outside. However, there is a simpler way. Draw a line from the point to the outside of the curve, crossing the curve as much as you like, and count how many crossings you get. You will find that no matter how you draw this line, the number of crossings differs from each other by an even number – they will either all be odd or all even – why is that? Which numbers of crossings indicate that the point is on the inside of the curve? The key observation for your child to discover is that each time you cross the curve you are transitioning between the inside and the outside. This makes a great exploration for a child to investigate who is just learning about even and odd numbers.
Game – Sprouts
The game of Sprouts is a fun way to practice with the insides and outsides of curves. The game starts with placing 2 or 3 dots anywhere you like on a piece of paper. From then on two kinds of moves for adding line segments are allowed – either the new line connects two different dots, or the line connects a dot to itself. A line may be curved, but it may not cross itself or any other line. Also, no dot may have more than three lines connecting to it (if the dot has a line that starts and ends at the dot, that is counted as two connections for that dot). When a line is added, a new dot is also added somewhere along the inside of the new line (not at the endpoints). The player who adds the last legal line wins.
Here is an example game of Sprouts starting with two dots. This is copied from the Wikipedia article (link given below).
For children that enjoy this game, there are lots of possible variations. There are also some interesting ways to analyze the game. As a reference, please check out the Wikipedia article on Sprouts.
Knots, Braids, and Links
There are many ways that a curve can be entangled with itself or with other curves. Some of them are quite beautiful and are forms of art in many parts of the world.
Knots in 3 and 4 Dimensions
A knot is a simple closed curve. If the pieces of a loop can be smoothly moved around so it looks like a standard circle, then it is called an unknot.
All closed loops on a piece of paper must be unknots. You may want to see whether closed curves can form knots on a torus (the surface of a donut) or other more complicated 2-dimensional surfaces.
All knots in four dimensions and higher can be untied using the extra dimensions, so all those knots are simply unknots. Play with this four dimensional idea with your child by thinking of the fourth dimension as time and looking at ways to untie a knot using the time dimension. Surprisingly, this one property of spaces with more than three dimensions is a crucial difference when topologists study spaces.
How can you tell when two knots are the same? How can you tell if the drawing of a knot is actually a drawing of an unknot? It’s not always easy even to tell whether a drawing shows a single loop or multiple loops.
Check out the Wikipedia article on Knot Theory if you would like to see descriptions of the basic knots. It also describes steps that can be taken to show that two drawings of the same knot are actually the same.
Two Puzzles – Loops and Connections
Solving either of these puzzles is helped by taking a topologist’s point of view.
This first puzzle is great as an icebreaker at a social event. Start by breaking people into pairs. Give each person a long piece of string (at least 4 feet long) and ask them to tie each end of the string loosely around their wrists (the looseness is important, but just tell them it is for their own comfort). The one catch is that while they are setting things up, their string needs to be looped through their partner’s loop, as shown in the drawing on the left.
The puzzle is to challenge them to get themselves apart without cutting the string. It’s important that the string be long enough so that people will be tempted to step through the other person’s loop and do other silly things.
As a topologist you know that if the string is attached tightly to the wrists, then the string and the body make a simple loop – and if two simple loops are connected, then there is no way to disconnect them without cutting. So, the loose connection at the wrist is essential. Next, again as a topologist, you know the amount of space around the wrist does not matter, so make it a huge space. Once you’ve done that, the solution will become apparent.
The second puzzle is fairly easy and involves two sets of three dots, using the same three colors in each set. The challenge is to draw three curved lines, each line connecting two dots of the same color, and no two lines crossing each other.
It doesn’t matter how you arrange the six dots, so you can play around with arrangements that may seem more challenging for your child. To help limit the possibilities, you can take the top three dots and attach them to the top of the frame, or you can take four of the dots (as shown) and attach them to the frame.
The key to seeing how to solve these puzzles is to realize that the dots not attached to the frame can be moved wherever you like (the topologist’s point of view again). So, move the dots to places where it is easy to solve the puzzle and draw in the lines – then, with the lines still connected, slowly move the dots to where they are supposed to be while bending the lines as needed.
Art – Loop and Knot Patterns
Many cultures around the world have done beautiful work with weaving loops and knots into intricate and interesting patterns. This article is already getting a little long, so I’ll just leave you with links for examples of the remaining material.
Celtic knot patterns: Start with Wikipedia on Celtic Knot.
Lusona (plural of Sona) sand drawings from Africa: Start with Wikipedia on Lusona.
Girih – Islamic knot patterns: Start with Wikipedia on Girih.
Macramé and tatting knots: Once again Wikipedia has an article on macramé and one on tatting, but you can also find a great many how-to sites showing you how to make the various knots.
Braids: Just as for knots, there is a whole mathematical theory of braids. A lot more enjoyable for the average person is all the articles you can find on hair braiding and braiding to make jewelry necklaces.
Wrapping Up (so to speak) I hope I’ve introduced you to some new ways to have fun playing with curves and loops with your child. Next month we will go up one dimension and play with surfaces.
If you have any questions or comments, please send them our way! We would enjoy the opportunity to chat with you. Also, if you are interested in collaborating with us or supporting us in any fashion, we would love to talk with you about ways we can work together!
June 18, 2024
Chris Wright
Chris@EarlyFamilyMath.org
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