The 17 Wallpaper Groups

“Wallpaper groups”? What does that mean anyway…

For the final project of my capstone class, my group and I decided to produce a tessellation coloring book. Since we are all future elementary math teachers, we thought this was a perfect, relaxing idea. Make random designs, generate them on a computer, bind them together, and BAM…a coloring book! However, our professor thought to challenge us and introduced us to the 17 Wallpaper Groups. wallpaper definition

Each of the 17 symmetry groups is made up of translations, rotations, reflections, and glide-reflections, or any combination in between. I started researching about each symmetry group so I could decide how to incorporate it into our project. There was a lot of information on each group, such as: lattice type, IUC notation, rotation orders, and reflection axes. Here, is a spreadsheet to organize my findings.

After looking at all of these properties, I started to wonder about the history of these specific groups. Who was/were the mastermind behind our symmetrical wallpaper?

A brief summary of events, as told by Natalie:

It turns out the 17 plane symmetry groups were discovered during the 19th century. A collection of people throughout mathematical history added to the development: first, the Pythagoreans discovered that there are five regular solids (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron). These were used for a variety of purposes, but shined in architecture and decorative art. A man more famous in astronomy, Johannes Kepler has influenced work in mathematics as well. He is generally credited for the first systematic explanation of the set of tessellations known as Semi-Regular or Archimedean. There are 11 specific examples of these tessellations which are composed of regular polygons.

Nimage014ext up, Robert Hooke looked at
different ways atoms could be arranged to form crystals. Then, in 1831, Hessel classified the 32 three-dimensional
point groups that correspond to the three-dimensional crystal classes. A crystal is an endless repetition of some point group operations in a 3D plane, or “wallpaper”. The wallpaper, or space lattice, is generated by tessellations of this symmetrical point group.

Now, in 1835, Frankenheim (possibly related to Frankenstein?) geometrically found all the symmetrical network of points that can have crystallographic symmetry.

A French mathematician by the name of Camille Jordan discussed a general method for defining all the possible ways of regular repeating identical groups of points (that’s a mouthful). Through this theory, he discovered 16 of the 17 wallpaper groups. It was not until 1891 when Fedorov, German mathematician Schonflies, and English geologists Barlow, proved that there were only 17 possible patterns in “The Symmetry of Regular Systems of Figures” (two-dimensional crystallography groups). And alas, the 17 Wallpaper Groups were born.

wallindex


 

The discovery of these groups dates back a long time, with mathematicians and scientists from all over adding their insight. This was rather interesting to learn about and will be extremely helpful to my project!

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Is Math a Science?!?

Is math a science? A question that seems so simple to answer, yet is far from it. To classify ‘mathematics’ as a branch of science seems a bit much to me. Yes, we use math to do science, and yes, we sometimes use science to do math. However, the two have certain distinctions that keep them separate. Based on research and my own thoughts, I would say that math and science are closely related, but not one in the same. Here is why…

The main difference between math and science: how ideas are tested and accepted.

In science, the task is to figure out what rules/laws are operating by observing the results following the rules. If your predictions conflict with your experimental data, then you need to change the set of rules you picked. Math is the opposite though: we choose the rules, with the task to discover what the results will be when choosing this particular set. Here, our choice in rules yields no specific right or wrong outcome. If the results produced are interesting enough, another mathematician will surely keep playing with them.

One of the criteria on the Science Checklist is relies on evidence. This relates to science, of course – evidence can make or break a scientific theory. Science NEEDS evidence to be accepted! And the more evidence, the better! However, there is always new evidence being discovered. Constant revision of theories means scientific ideas can never be absolutely proven. Again, math doesn’t work this way. Mathematics relies on proof. No proof? No deductive reasoning? Shot down! Not accepted!

Doctor Ian, from mathforum.org, summarizes his response to the question quite nicely:

“Science is the pursuit of the correct description of this particular world; whereas math is the pursuit of interesting descriptions of possible worlds. Whereas scientific theories are right or wrong, mathematical ‘theories’ are merely interesting or uninteresting.” – Doctor Ian


References:

Book Review: How Not to Be Wrong

I recently read How Not to Be Wrong: The Power of Mathematical Thinking by Jordan Ellenberg. My feelings about the book roller-coastered until the last page. Overall, I did enjoy this read.

One afternoon this summer, I was exploring Holland with my sister in town. We walked into multiple shops and boutiques, but of course found ourselves lost in a bookstore on the corner. Here, was my first encounter with Ellenberg’s book. As soon as I walked in, How Not to Be Wrong was featured in the front; the red dart and word “mathematics” on the cover caught my eye. I read the back, flipped through a few pages, and took a picture of the cover so I could read it in the future. Lucky for me, this book was on the recommended book list for my capstone class, so of course I jumped right on it.

Initially, I loved this book. I loved it so much I hardly put it down – I even got splashed in the face by a sprinkler while I was reading and walking to class…not okay! Ellenberg begins the book with a perfect explanation to a high school student about what mathematics is. I quoted a majority of it in my blog post, What is Math?  He states: “Math is woven into the way we reason…it is the study of things that come out a certain way because there is no other way they could possibly be.” And then dives into the story of Abraham Wald, a mathematician asked to analyze bullet-hole data from planes returning from World War II sorties. Wald shocked the military when he suggested putting more armor where you don’t see the bullet holes. The reason: planes hit in such spots did not return to be included in the data. This is the kind of “mathematical thinking” Ellenberg highlights throughout the rest of the book, “the extension of common sense by other means.”

After finishing the first couple chapters, my feelings began to develop. The book is divided into five major mathematical topics, each with one or two chapters on specific events/stories to help better understand that topic in general. Here is where my loss of interest started. I noticed myself going back a line, a paragraph, or sometimes an entire page, just to make sense of what I read. His stories are relevant, but fail to the overall connection: when am I going to use this? Ellenberg points out complex mathematical ideas from our everyday lives, and tries to simplify them for anyone with a high school education to understand. His result: a long, confusing chapter, over-explaining what he is actually trying to get across.

Although Ellenberg takes on a lot to chew here, he does make it a point to keep his voice active throughout the entire book. I caught myself laughing out loud multiple times while reading, something that really helped in finishing the book. Even though I had to go back three times to read the same passage, I didn’t mind if the description of the +/- sign in the quadratic formula “looks like a plus sign and a minus sign that love each other very much”.

In conclusion, I would recommend this book to anyone who truly loves mathematics. The stories are in fact eye-opening, only once you understand the point he is trying to make. I think Ellenberg just needed to connect his final thoughts on each story to his main point of the book: mathematics is everywhere and anyone with some mathematical background can see that.

Thales, and Pythagoras, and Euclid…OH MY!

As much as I love math, I’ve realized how little I know about the actual history of it. Completing the square, proving conjectures, and using basic mathematical operations, are concepts that I developed an understanding for. But where did it all come from? Who took the time to figure all this ‘math stuff’ out?

Focusing on Greek mathematics, I began to do some research and was intrigued with my discoveries. (*Note: I will post references at the end of the blog post.)

I found this 2-minute video that gives a nice overview on how Greek mathematics influenced our society today. Watch it here!

Thales, Pythagoras, and Euclid are some of the most well-known mathematicians – not just in ancient Greece, but today too! These three, along with countless others, paved the building blocks for mathematics as we now know it. The Greeks insights/discoveries presided over one of the most dramatic and important revolutions in mathematical thought of all time.

Though they dabbled a bit with number systems and algebra, most of Greek mathematics was based on geometry. Thales, one of the Seven Thales-Quotes-3Sages of Ancient Greece, is considered to have been the first to lay down guidelines for the abstract development of geometry – using deductive reasoning first, mind you. He tried to establish axioms, or statements that could be accepted as truths, and thought that all other mathematical laws could be deduced from them.

Thales’ four main mathematical proofs were:

  1. The diameter of a circle exactly bisects the circle
  2. The base angles of an isosceles triangle are equal
  3. The two pairs of angles formed by two intersecting lines are identical
  4. If one side and the two adjacent angles of a triangle are shared by another triangle, the triangles are identical.

Although they seem extremely easy and obviously correct to us math majors, these first principles fueled an explosion in the study of mathematics, and taught many of the mathematicians that would follow him and build upon his theories (30 years later for Pythagoras and 300 years later was Euclid). Such basic ideas nowadays that came from years of dedication and exploration; it is shocking how long it took these concepts to come about!

Next on our list: Pythagoras.

44_pythagorean_theoremThe Pythagorean Theorem: one of the best known mathematical theorems of all time. I can bet (only hope, really) that a majority of adults can recite this theorem correctly. The theorem has to be the most proven of any mathematical theorem around!

Pythagoras was perhaps the first to realize that a complete system of mathematics could be constructed, where geometric elements corresponded with numbers. Like Thales, he worked upon proving axioms and using these to deduce other mathematical laws, building theorems upon theorems. Pythagoras, along with his Pythagorean friends in a secret society, developed the concept of trigonometry and left us with an understanding of shapes, angles, triangles, polygons and proportions that we continue to use today.

Last the list: Euclid, a very famous name among all mathematicians and is often referred to as the father of geometry. Gathering the work of all of the earlier mathematicians, Euclid created his landmark work, The Elements, which is said to be one of the most influential non-religious books of all time. Here, he established 10 axioms (also referred to as ‘postulates’). Some of these axioms seem self-explanatory, but Euclid operated on the principle that no axiom could be accepted without proof. Euclid’s postulates are shown below:

  1. Things which are equal to the same thing are also equal to each other.
  2. If equals are added to equals, the results are equal.
  3. If equals are subtracted from equals, the remainders are equal.
  4. Things that coincide with each other are equal to each other.
  5. The whole is greater than the part.
  6. A straight line can be drawn between any two points.
  7. Any finite straight line can be extended indefinitely in a straight line.
  8. For any line segment, it is possible to draw a circle using the segment as the radius and one end point as the center.
  9. All right angles are congruent (the same).
  10. If a straight line falling across two other straight lines results in the sum of the angles on the same side less than two right angles, then the two straight lines, if extended indefinitely, meet on the same side as the side where the angle sums are less than two right angles.

Seems pretty simple, right? Not exactly. Euclid used these 10 postulates to develop 465 propositions, progressing from established principles to the unknown.

The reason that Euclid was so influential is that his work was much more than just an explanation of geometry and math related topics. Euclid provided logical arguments and demanded proof for every theorem that crossed his path. These concepts of proof, mathematical rigor, and deductive reasoning gave mathematics its power and ensures that proven theories are as true today as they were thousands of years ago. Personally, I think this is one of the most important contributions of Greek mathematics.

To be continued another day; thanks for reading! (:


References:

What is Math?

Another year has begun! This means another year of blogging. I am excited to start using this again, and hope to blend it into my daily routine.

This is my last semester before working hands-on in the classroom. A semester packed full of math: Euclidean geometry, modern algebra, statistics/probability, and a history of mathematics capstone. With math being my emphasis, this doesn’t seem too daunting.

On the first day of my capstone course (MTH 495), we discussed two important questions: what is mathematics, and what was the first mathematics? You would think a room full of math majors could answer this without hesitation, but that wasn’t the case. Some of our responses are below:

  • A numerical/logical explanation of the world
  • A mix of physical and abstract concepts
  • The science/study of patterns
  • The first mathematics coming from: measuring, counting, trade, structures, distances, cooking, and comparisons.

I agreed with my classmates about the presented ideas. However, when reading my chosen book for the course, some passages truly stood out to me about the question “What is Math?”. From Jordan Ellenberg’s How Not to Be Wrong: The Power of Mathematical Thinking, 

“Math is woven into the way we reason. And math makes you better at things. Knowing mathematics is like wearing a pair of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world. Math is a science of not being wrong about things, its techniques and habits hammered out by centuries of hard work and argument. With the tools of mathematics in hand, you can understand the world in a deeper, sounder, and more meaningful way.”

“We tend to teach mathematics as a long list of rules. You learn them in order and you have to obey them, because if you don’t obey them you get a C-. This is not mathematics. Mathematics is the study of things that come out a certain way because there is no other way they could possibly be”.

As a future teacher, these explanations of mathematics seem perfect to share with students. Mathematics has grown to have such a negative connotation that students need to be shown the importance of math in our every day lives.

I am excited to explore the history of mathematics, and plan to add my future discoveries to this post! Happy Blogging (:

Teaching Portfolio

This is a collection of resources related to the mean as a measure of center, and the Mean Absolute Deviation as a measure of variability. My MTH 323 class (Probability and Statistics for Middle School Educators) joined with a middle school teacher to teach a couple lessons on these topics. While most of the class got to teach a few students with a partner, my group was able to go in early and observe their teacher – and student teacher – give the lesson. We contributed our ideas in the planning phase of the lesson, and watched to see how it went with the students.

Here are some of the lessons we used, as well as additional resources on these concepts!

Lesson1(JH)_Balance PointLesson_Outline

Lesson2(JH)_MADLesson_Outline

Lesson3 (GA activity)

Mean: Fair-share & Balance point with Middle School students!

Unfortunately, I didn’t get the information regarding a blog post about our learning focus until after we taught the lesson. However, I still would like to reflect on the experience since I learned a lot in just one hour!

As a teacher, we make mistakes. We are not perfect and cannot claim that we are. Things will happen, and we need to brush them off and continue to grow from them. Working with the students in Zeeland last week was extremely beneficial to me. I had the opportunity to watch a phenomenal woman (and her student teacher) interact with her students and teach a lesson my class helped create. Tara truly knows what she’s doing up there, inspirational for any teacher. So, simply observing how she runs her classroom is impressive. My partner and I, along with our Professor, the student teacher, and Tara, walked around to each group to help answer questions. Here was where I messed up. I told the first group to stop me to make note of the two frames of bowling, making them think the mean was 8 instead of the correct 16. My Professor approached me explaining the difference. I felt so embarrassed! I flipped through Tara’s lesson the day before, but obviously not as well as I should have. If I plan to help these students, I need to fully understand what is being taught and be ready for any questions they throw at me. Know the difference between two similar approaches. A list of anticipated questions. Not things I emphasized the first lesson, but something I will take to my teaching grave!

Overall, this first lesson went well. Some bumps along the way, but things I will make note of and improve for next time. I am excited to be back in Tara’s classroom a couple more times this semester!