GMU Math MakerLab: September 2019

Friday, September 27, 2019

Pentagon Tiles Math 401 Fall 2019

Pentagon Tilings

Math 401 Mathematics Through 3D Printing Fall 2019

Students in Math 401 have printed 13 of the 15 pentagon types that tile the plane!

These 3D printed objects are designed and printed by students in Math 401, Fall 2019, taught by Instructor Dr. Evelyn Sander with the assistance of Chloe Ham and Colin Chung.

15 classes of irregular pentagons tile the plane

The fact that there are exactly 15 distinct classes is still under peer review. These tilings represent examples of a subset of the 15 classes.

Wednesday, September 25, 2019

Pgg Wallpaper group by Connie Quezada

The following post is by Connie Quezada as a part of George Mason University, Math 401, Mathematics through 3D Printing.

Explanation of wallpaper groups:

A Wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on symmetries in the pattern. These patters can be found frequently in architecture and decorative art.

Evgraf Fedorov proved that there are only seventeen distinct groups of possible patterns in 1891.

The symmetry of a pattern is the way of transforming the pattern so that it looks exactly the same after the transformation. Examples of symmetries are translation, reflections, and rotations.

The notation for groups begin with either with p or c for primitive cell or a face-centered cell, then followed by a digit indicating the highest order of rotational symmetry, and the next two symbols indicate symmetries relative to one translation axis of the pattern and are either m for mirror, g for glide reflection, or 1 for none.

My Wallpaper group:

The group pgg contains two rotation centers of order two (180), and glide reflections in two perpendicular directions. There are no reflections and the centers of rotation are not located on the glide reflection axes.

This wallpaper group can be found on Mesopotamian artifacts and is also used as a way of tiling bricks and tiles on roads.

On my print the different colors are used to illustrate the glide reflections being performed on a single tile. When I printed, I printed the original tile and the reflection of the tile with a star design placed on top.

Sources:

https://en.wikipedia.org/wiki/Wallpaper_group#Group_p2gg

Wallpaper Patterns: P1

by Khalilah Hamer

For the week of september 16th 2019, the class of Math 401: Mathematics Through 3D Printing, was to demonstrate different wallpaper groups. Wallpaper patterns have been dubbed that for the fact that these patterns look like wallpaper and how they tile the plane as well as wallpaper covering the wall. We see these patterns occur frequently everyday whether it be art, floor tiles, and other architecture.

(https://www.youtube.com/watch?v=5UbMFiK3LY0)

In 1891, Russian mathematician, Evgraf Fedorov proved that only 17 distinct wallpaper patterns exist. In these patterns there are 3 different transformations that can occur. A shift or slide may occur when the tile/pattern is shifted producing the same tile as started with, this is a translation. A turn or rotation can occur where a tile is rotated around a certain axis leaving the new tile to be rotated a certain degree from the original. The last transformation that can happen is a flip or reflection where a tile is reflected in a direction and the resulting tile is a mirror of the original. There is a special case where a tile may have a glide reflection where a tile is shifted then reflected across an axis. Examples are shown below.

https://www.onlinemathlearning.com/reflection-rotation.html

Everyone in the class was given one specific tiling group to create their own pattern out of. I was given the task of demonstrating group p1 which is only contains translations, no rotations, reflections, or glide reflections. Although it seems like an easy pattern to follow, when designing the pattern I kept creating shapes that would have a reflection or rotation. Until I reached my final design.

(https://en.wikipedia.org/wiki/Wallpaper_group#/media/File:Wallpaper_group-p1-3.jpg)

I created a “juicebox” shape and I realized it was a simple hexagon that could be reflected so to combat that I added a “straw” to the design to not get it confused. Adding the straw was the hardest part as I had to find measurements. It took me a bit to create the exact measurements to later find out that when printing there is a bit of an error in printing which I explained in the last paragraph. As shown in my print, the “juice box edges” aren’t printed but showed up in the stl file. As someone new to 3D printing I am not sure what happened there.

If I were to do this project again, there were a few mistakes I had learned.First, I needed to add a bit of a buffer room for the “straw” piece, as they were not fitting together at first. As I printed on Friday, I didn’t have much time to reprint so I did take an exacto knife to it and shave off a bit of the print until I could connect the pieces together. It ended up working, but I know for next time to leave the space. I would also see what happened to the “edges” of the print as they didn’t show up on the pieces. If they did it might help to see the whole picture. Without the mistakes I wouldn’t learn to do better for the next print.

Pentagonal Tiling of a Plane (Type 5)

Joseph Kerns
M401
Assignment 1

Throughout history, artists and architects have attempted to make patterns fit to whatever surface they have available. Typically, there is beauty in patterns and repetition. Within those patterns, mathematicians have been able to work out some, but not all of the methods used to tile a plane. I’ll focus on just one such tiling, which has the following rules:

1. Only one shape can be used
2. No gaps between the shapes
3. The shape must be convex

With these rules in mind, let’s look at pentagons. In 1918 Reinhardt proved classifications for the first five types of pentagonal tilings of a plane. But, the pentagons didn’t stop there. 50 years later, some progress was made regarding the different tilings of the plane. All in all, there has been 15 types of pentagons that have been shown to tile a plane monohedrally. This may not be the complete list, but serves as the most complete that we have today.

For my project, I was tasked with creating a pentagonal tiling of Reinhardt’s Type 5 pentagons. The rules for this pentagon are pretty simple:

1. One angle must be 60 degrees
2. One angle must be 120 degrees
3. The line segments that create the aforementioned angles must be equal length.

This tiling can be oriented in two separate ways to cover an entire plane. Either the tiling has a unit composed of 6 or 18 pentagons, as shown below.

Utilizing OpenSCAD software, I modified the angles of the pentagon that deviates from the basic case by increasing the size of one of the angles. Below is the code and an image of the shape, along with a 3D printed version that is oriented in the tiling it creates.

Source: https://en.wikipedia.org/wiki/Pentagonal_tiling

Tiling of the Plane With A Tree

Illia Stadnyk

Wallpapers have a unique patterns where they can be formed using the same pattern over and over again. The shapes are usually constructed using one shape but can be done with more than one shape as long as the follow that they have no overlap and no gaps, such as simple walkway using squares. In mathematics this is done usually using translational symmetry and rotations, but in the end the patterns share a symmetry.

These tiling shares a lot with crystals who they share patterns with. That the rotations are only in groups of 2, 3, 4, or 6. That the symmetry in the crystal is that of the tiling groups and is referred to as plane crystallographic group. They come in total of unique 17 groups. Each group has a name that refers to the number of rotations, if it has a mirror or a glide reflection, and if it is primitive cell or centered cell. The proof for these 17 groups was written by Evgraf Fedorov in 1891.

The group we will look today is pg, or the full name being p1g1. It is a primitive cell that has the highest order of rotation of 1, and that it has a single glide rotation. It is relatively simple wallpaper, the pattern can be seen through a lot of patterns such two rectangles that make the end of odd looking arrow. It can be made with many shapes as long as they can be fitted in as there is no rotations needed.

To make create wallpaper group pg can be most easily done with a square or with a triangle as they can easily be used for tiling. A more interesting is combining these shapes to create is something that resembles a Christmas tree. To make sure it works properly the shape has to be made so that glided reflection has no gaps, if that is satisfied then it will be a pg group.

In my case the shape was similar that of a basic tree. The design originally did not have correct spacing so the shape can an opening where the bottom parts of the glided reflection was supposed to connect. Simple extension from a simple triangle shape to more of a rectangle shape allowed the extension to make sure there was no gap bellow.

Citations:

https://www.fiz-karlsruhe.de/sites/default/files/FIZ/Dokumente/Broschueren/A-Focus-on-Crystallography.pdf

http://mathworld.wolfram.com/WallpaperGroups.html

https://en.wikipedia.org/wiki/Wallpaper_group#Group_pg

Tuesday, September 24, 2019

Thingiverse Entries, Math 401 Mathematics Through 3D Printing, Fall 2019

Wednesday, September 18, 2019

Pentagon Tiling Type 2

The following post is by Katharine Downey as a part of George Mason University Math 401, Mathematics Through 3D Printing.

Pentagon Tiling Type 2 – by Katharine Downey

A pentagon tiling is a tiling of a plane in which there are no gaps between any of the pentagons, however all pentagons must be of the same shape.

This is an image of the pentagon tiling I printed for MATH 401, Mathematics Through 3D Printing at George Mason University.

Brief History and Background:

This pattern type is a Type 2 and was found by Reinhardt in 1918. He was the founder of the first five types of pentagonal tilings. There are fifteen types of convex pentagons today that are known to tile a plane monohedrally, the latest of which was found in 2015.

The history of the pentagon tiling discoveries picks up after Rinhardt mush later when Kershner discovered the next three in 1968 who incorrectly stated he had completed the list. Type ten is credited to James in 1975 and shortly after in 1977, a stay-at-home housewife, Marjorie Rice, with a fascination for mathematics found four more. The fourteenth was found some time later in 1985. A fifteenth was found in October of 2015 by Mann, McLoud, and Von Derau from computer algorithm.

Michaël Rao showed that this list of fifteen tilings was complete in that there are no other possible tiling types of pentagon that can tile a plane without any gaps in 2017.

All five types that Reinhardt found can create isohedral tilings, meaning any tile can be mapped to any other tile due to the symmetries. A more formal mathematical explanation of an isohedral tiling is that the automorphism group acts transitively on the tiles.

B. Grünbaum and G. C. Shephard have shown that there are exactly twenty-four distinct types of isohedral tilings of the plane by pentagons according to their own classification scheme. All of these twenty-four types use Reinhardt's tiles, usually with additional conditions necessary for the tiling. There are two tilings by all type two tiles, and nine of the twenty-four tilings are edge-to-edge variations. There are also 2-Isohedral tilings by special cases of type one, type two, and type four tiles, and 3-Isohedral tilings, all edge-to-edge by special cases of type one tiles.

My Tile:

I created my tiling in a program called OpenSCAD, while it is mainly used as an imaging software for 3D printers, it is a very powerful imaging software for geometry and other purposes. After working out the angle and side measurements, I placed a few pentagons in order to see if I calculaded (or coded) the tiling correctly. As some may know from programming/coding various things, a code will hardly ever work the first time which is why I left the angle/side modifiers in. Then I modified the angles or sides in order for the tiling to monohedrally tile the plane.

Here is a piece of the code I used to create my pentagon in OpenSCAD:

This is a piece of the code, not including the angle and side measurements nor the angle/side modifiers. I wrote in order to print out ten type two tiling pentagons.

This is an image of a four-tile primitive unit with one of the tiles out of place to help visualize the shape of the tile.

The angle and side requirements for a type two tile are:

c = e and B + D = 180 degrees

My angle measurements for my tiles are as follows:

Angles: B + D = 180, A = 88, C = 140, E = 132

The wallpaper group symmetry for my printed tiling is pgg (22x), with orbifold notation in parentheses. The wallpaper group for all type two tilings are pgg (22x) unless the mirror image protile tiles are consided distinct, then the symmetry is p2 (2222).

Here are some other Type 2 examples:

The first is pgg (22x) and the second is an edge-to -edge variation with p2 (2222) symmetry.

Sources:

https://en.wikipedia.org/wiki/Pentagonal_tiling

https://www.quantamagazine.org/marjorie-rices-secret-pentagons-20170711/

https://sinews.siam.org/About-the-Author/pent-up-using-pentagons-to-tile-a-plane

Monday, September 16, 2019

Rice’s Tessellating Pentagons: Type 11

Taylor Fountain

Math 401

09/16/2019

Assignment 1

Printed Type 11 Pentagon

Throughout the exploration of tessellations, pentagons have remained a point of interest as the lowest-order polygon whose regular variant cannot tile the Euclidean plane. This is due to the internal angles not being a divisor of 360°, preventing a full rotation from being formed without gaps or overlaps between them. With regular pentagons out of the picture (or the plane), the search for irregular pentagons that tiled the Euclidean plane commenced, resulting in the discovery of 15 distinct types convex pentagons that monohedrally fulfill the task.

Between 1976 and 1977, amateur mathematician Marjorie Rice discovered four new types of pentagons that tessellated the Euclidean plane. All 4 were 2-isohedral and demonstrate pgg symmetry (1 of the 17 types of wallpaper patterns), which is characterized by two distinct centers of 180° rotation and glide reflectional symmetry. Type 11 adhered to the constraint equations seen below (see left diagram for labeling of sides and angles), and consisted of an 8-tile primitive unit (see right diagram).

The code for the Type 11 Pentagon presented with only one modifier, which was of angle AA (angle E in our diagram). Setting the angle at a value greater than 130° or less than 110° resulted in quadrilateral and triangle that met at a point, so the range for possible values was fairly small. In the end, I set the angle to a value of 120°.

Code in OpenScad

I printed my extruded pentagon on the MakerBot printer. One error I encountered was the placing of the 8 tiles significantly increasing the time to print: by spacing out the tiles on the plate, the extruder had to lift several times when forming the base, whereas a more condensed arrangement of the tiles may have decreased the required print time. In all, the print took 2 hours and 31 minutes, so, while there was no print slot directly after mine, I should consider placing separate objects closer together in the future in order to decrease print time. In short: think more like a computer.

8-tile primitive group, exploded for clarity

An idea that occurred to me while viewing the primitive unit was that, when the adjacent pentagons that were reflected over side b were view as one piece, the unit could be comprised of four identical heptagons, each with one concave angle. Using this shift, some conclusions could be drawn about tiling the Euclidean plane with irregular heptagons containing concave angles.

Monday, September 9, 2019

Linear Transformation of a Sphere

Michael Tritle

09/09/2019

MATH 401: 3D Printing

Assignment 0

Consider two vectors u,v in V and a scalar q in F, where V is a vector space, and F is a field. A transformation is called linear if it satisfies two properties.

Intuitively and geometrically, we tend to think of linear transformations as rotations and shifts while keeping grid lines intact. If we were to consider a Cartesian Plane of real numbers, we see it like this:

T(u)

Now let’s consider a sphere that lives in 3D real space. There are a few different ways to represent a sphere in 3 dimensions. We can consider the set of all points that is equidistant from the origin. Using the distance formula in 3 dimensions we achieve

where k is an arbitrary real valued constant. Let to apply a linear transformation to a sphere, it would be easier to work with this in vector form. Using spherical coordinates, it’s a straight forward procedure:

We’d now like to find out to find a basis for R3 that will represent the boundaries of our sphere of radius 1. Letting theta = pi/2 and phi = 0 results in (1,0,0), letting theta = pi/2 and phi = pi/2 results (0,1,0) in and letting and letting theta = 0 and phi = pi/2 results in (0,0,1).

Let’s now define a linear transformation that scales the boundaries of our sphere to and. Such a transformation can be represented by the matrix

Left multiplication on A will transform our vector representation of a sphere as wanted;

If we write this as an implicit equation, we find that

This is an equation for an ellipsoid! So the linear transformation we defined mapped the coordinates of our sphere to the coordinates of an ellipsoid. It might be an interesting exercise to see if all non trivial linear transformations from on an sphere are mapped to an ellipsoid.

Creating a 3D printable representation of this ellipsoid can be made in openSCAD by scaling a sphere by it’s components. The code used for the print is:

scale([3.0,2.0,1.0]) sphere(r=5.0);

Below are some pictures of the ellipsoid as a 3D model, and as a 3D print.

The Symmetry in a Basic Temple

The Symmetry in a Basic Temple by Cassandra Hulbert

MATH 401

9 September 2019

About My Print:I generated my inspiration for this project from my recent trip to Tokyo, Japan. The temples within Japan date as far back as the early 7^thcentury and are found in abundance throughout the country. Early construction centered around nature, aesthetic, and symmetry. I wanted a three story pagoda, with each floor slightly smaller than the previous and a characterizing lightning rod. The idea was simple enough, but required a composite of different techniques within OpenSCAD to create. The object puts to work several of the techniques learned in week 1 tutorials.

Creation Process:With the use of entirely cubes and cylinders, this object was able to be stacked and formed. Starting on the lower level, a cube of [20,20,6] measurements was developed, with a difference cube of [20,5,3] to cut out a doorway. To top the “main room”, I developed a cylinder which was pinched at the top, flattened, then adjusted to 4 facets, ($fn = 4), to create a square-based with a sliced top, also known as a frustum. I repeated this process of a cube topped with a frustum, with each overhead layer having smaller dimensions (of -5 in length and width), until I got to base layer 3. The cylinder which made up the rod, with a radius of only .5, was of relatively small-scale dimensions. The hanging edges in the rendering of my code proved to be a slight problem when it came to print time. Nevertheless, my object was now ready to be printed.

Printing Process:The Ultimaker Cura put my print time at roughly 2 hours. About 45 minutes in, the plastic starts to dip in where the edges hung. I had to stop my print, add supports in, and re-print. For it being my first print, I expected error, hoped for perfection. With the supports in and a smaller scale, my print was able to fit in the allotted timeframe. Ultimately, I was pleased with the quality, accuracy, and sturdiness of my final product.

Spiky Star Wallpaper Pattern (*442)

Spiky Star Wallpaper Pattern (*442)
Savannah Crawford - MATH 401 F19

Over the summer I went to a summer school program in Vancouver, BC that was hosted by the Pacific Institute for the Mathematical Sciences. One of the courses in the program was on symmetries, specifically the 17 wallpaper groups for tiling the Euclidean plane. So when Sander announced the topic for the assignment, this seemed like a natural fit for a project on symmetries.

In 1891, Evgraf Fedorov proved that there are 17 wallpaper groups which tile the Euclidean plane. The 17 groups are classified by their symmetries: mirror lines, points of rotation, and transitional symmetry. Using this approach, you can reduce seemingly different patterns to their symmetries and see that they are the same geometrically. Additionally, there's a cost function associated with each group, and the cost for all 17 groups is 2.

[This pattern belongs to *442 since there are two pints where 4 mirror lines intersect and one where 2 mirror lines intersect.]

My pattern is in *442 (Conway notation). It's one of the simplest groups. Graph paper is another example of *442, but my base shape is not a square so the pattern looks pretty interesting despite its group. Essentially, the base shape of my print is a spiky ring.

[Here is the spiky ring that generated the wallpaper.]

I tiled the base shape such that the edges overlap creating a grid with spikes. I left the edges round to showcase the geometry the original base shape.

[Final object to print]