Wallpaper Patterns on Tapestries

I’ve always been obsessed with tapestries whether they are authentic fabrics from other countries or just ones that you can buy off of Amazon. I have a black and white tapestry in my room above my bed that I just love, and it was on sale so I had to get it.

In the picture above is the design in the center of my tapestry.  It is dihedral and can rotate around the center 16 times from what I can count here, because I wasn’t sitting there counting the pattern in my room in my free time.

As you can see here, there is rotational, horizontal, and vertical symmetry here.  This is the pattern surrounding the main design on the inside.  It repeats over and over again all throughout the outside.

If you’d like to learn more about how tapestries are made, I’d advise you to check out this website. Enjoy!


Nature Mandalas

If you know me, you know that I’m a nature freak.  I always strive to be one with the Earth and feel the energies radiating off of its surface and all that stuff, I just find it so calming and interesting and it keeps me in the present. So obviously, I had to incorporate this into my blog post this week.

Natural mandalas can occur in so many places in nature, whether its in flowers, shells, spiderwebs, or even fruits.

Fruits hide a natural mandala in their core.  When it comes to citrus fruits, there are mandalas all throughout, but when it comes to fruits like apples, a perfect mandala is always formed right at the center.

Flowers can have a circular center blooming out to colorful outer rings that have layers of color.  Ferns can also be considered nature mandalas as well because the dewdrops on the leaves create symmetrical patterns.

Shells contain multicolored spirals throughout their entirety.  Spiderwebs, which I find the most intriguing and beautiful, create mandalas with a circular pattern beginning in the middle and working its way outward.  What makes them extra eye-catching is when the dewdrops (I don’t know why I’m focusing on dewdrops here but I guess I am) are glistening in the sunlight.  It just adds something extra to the whole image.

I could go on and on about this but instead, you can check out this link:


If you’re as into this stuff as I am, I recommend checking out the rest of the website while you’re at it!

Playing Around with the Four Color Theorem

First of all, I wanna give a quick shoutout to this dope snow storm that made me leave my house and go to Barnes and Noble just to have WiFi to post this. You rock!!!!!!

Anyways, the only thing I wanted to do today was play around with this four color theorem.  So… I researched some different aspects about it to widen my horizons.  We all know that the four color theorem is the idea that you can color in any reasonable map on a plane with just four distinct colors.  But how was this discovered and what happened in order for us to come to that conclusion?

In 1852, the theorem was brought up as a possible answer to this question of whether or not this was actually possible, but was not proven until over a CENTURY later in 1976 when it was the first major theorem to be proved using a computer.  How cool is that?

At this point, the people that were proving this theorem asked themselves if they could go any lower than four colors, so they tried three.  Many maps they experimented with showed success with this, since they started out just using maps of regions of the world… until they got to Europe.  If you look at the countries surrounding Austria, you’ll see that it is surrounded by a ring of seven countries, which clearly poses a problem because it’s not an even number, as shown in the picture below.

However, with the entire map of Europe, the four color theorem still holds its own.

In conclusion, my heart is still with the four color theorem.  Hope you all stay warm in this storm and hopefully not as butthurt as I am about having to leave my house!! If you’d like to read more about this theorem, please check out the link below:


Social Network Analysis

When it comes to graph theory, I was trying to find different maps that show trails or streets that I could further describe in detail because that was all I was thinking of to explore while considering the topic of this blog post.  Upon further research, I found that graph theory is not limited to maps and roads by any means, it can be found in a variety of examples using things we use in our everyday lives.

I researched multiple different real life applications of graph theory on a list on this website (if you would like to see more real life examples that I don’t discuss you can find it here):


However, the most interesting example I found was regarding social media.  The website describes the application of social media to graph theory as: “Connecting with friends on social media, where each user is a vertex, and when users connect they create an edge.” This sounded like a topic that was worth diving a little deeper into, since social media plays such a huge role in our lives.

I paraphrased that sentence and found that this real life application to graph theory is called “Social Network Analysis” or SNA.  This is described as being the mapping of relationships between people via different social networking websites, or different inanimate objects like computers, or even URLs.  The nodes in the network are defined the people in the groups, while the links between them show a mathematically visual flow of their relationships.  When it comes to computers, the nodes would be the computers themselves and the links would be the invisible connection each computer or smart device has when communicating signals back and forth.  Below, you will find a visual that shows an example of a Social Networking Analysis:

This network above in particular is known as a “Kite Network”, which was developed by a researcher named David Krackhardt. On the side bar to the right in the image, you can see the Degrees, Betweenness and Closeness information.  These are defined as the three different “Centralities” of the image.

The Degree Centrality is the number of direct connections each node (or person) has.  As you can see in the image, Diane seems to have the most connections.

The Betweenness Centrality is all about having few direct connections, but having the nodes you are connected with be some of the most popular.  In the image, Heather has very few connections, but she is between two very important nodes in the network.

The Closeness Centrality is having very short paths to others, like Fernando and Garth in the image above, but the pattern of their ties allows them to connect with everyone in the network anyways so they can be closer with fewer nodes.

Although this network seems to be neither a circuit nor a path as far as i can tell, (because of the line protruding from the side of the shape, slightly throwing it off) all of the connections are still meaningful because they are between people who created relationships with others and are making more relationships because of the never-ending flow of the social media network.

If you would like to learn more information on this topic, you can visit this website:


Facts About Tax

We all want to be millionaires, right?  At least I know I do.  What I didn’t know is that most millionaires live in the state of California, and that about 40,000 Californians report over a million dollars in income, compared to our neighboring state Vermont who has the least millionaires: only about 300 people in that state are millionaires.  In addition, more than 11,000 individual tax returns reported adjusted gross income above $10 million.  That’s a crazy goal to hit.  With an income of about $10 million, you could live in a house like this:

Pretty cool huh?

Another crazy fact is that our tax code has about 4 million words, and since about 17 years ago, there have been over four thousand changes to that tax code.

We all want to be a part of the top 1% of earners in the country, but you’ll need $369,509 of annual income to be a part of that group. If you’re shooting to be within the top 10% of earners in the country, you’ll only need a little less than half of that, about $116,555.

Have you ever wondered just how many people file their tax returns?  Turns out, the IRS receives over 140 million tax returns and collects just shy of one trillion dollars in taxes.

Want to find out more facts? Check out the link below: