- On 2010-11-12
- By Edugain

## Tessellations

A tessellation is simply is a set of figures that can cover a flat surface leaving no gaps. To explain it in simpler terms – consider the floor of your house. That is a flat surface – called a “plane” in mathematical terms. And you’ll notice that the floor is covered with some tiles or marbles of different shapes. That is a good example of a “tessellation”. The one difference here is that technically a plane is infinite in length and width so it’s like a floor that goes on forever.

Of course, when we are talking about floors, the shapes used to cover it are mostly rectangles or squares (in fact, the word “*tessellation*” comes from the Latin word tessella – which means “*small squar*e”). The word “Tiling” is also commonly used to refer to “tessellations”.

There are different kinds of tessellations – the ones of most interest are tessellations created using polygons. If you use only one kind of polygon to tile the entire plane – that’s called a “Regular Tessellation”

**Regular Tessellations**

As it turns out, there are only three possible polygons that can be used here. There are only three rules to be followed when doing a “regular tessellation” of a plane

- The tessellation must cover a plane (or an infinite floor) without any gaps or any overlaps.
- All the tiles must be the same shape and size and must be regular polygons (that means all sides are the same length)
- Each vertex (the points where the corners of the tiles meet) should look the same

Of course, you would have guessed that one is a square. What are the other two? They are triangles and hexagons. Let me show you examples of these two here.

You may wonder why other shapes won’t work. Let’s try with pentagons and see what shape we come up with. You can see that there is a gap and that’s not allowed.

So what’s unique to those 3 shapes (triangle, square and hexagon)? As it turns out, the key here is that the internal angles of each of these three is an exact divisor of 360 (internal angle of triangle is 60, that of square is 90, and for a hexagon is 120). The mathematics to explain this is a little complicated, so we won’t look at it here

**Semi-Regular Tessellations**

If you use a combination of more than one regular polygon to tile the plane, then it’s called a “semi-regular” tessellation. If you look at the rules above, only rule 2 changes slightly for semi-regular tessellations. All the other rules are still the same.

For example, you can use a combination of triangles and hexagons as follows to create a semi-regular tessellation. There are eight such tessellations possible

There are many other types of tessellations, like edge-to-edge tessellation (where the only condition is that adjacent tiles should share sides fully, not partially), and Penrose tilings. Each of these has many fascinating properties which mathematicians are continuing to study even today. Tessellations are also used in computer graphics where objects to be shown on screen are broken up like tessellations so that the computer can easily draw it on the monitor screen.

superb i like your website

It is good too read your website again!, i see some interesting updates here..

What a great resource!

What a great site. Superb.........!

An attention-grabbing discussion is price comment. I believe that you need to write more on this subject, it won't be a taboo topic but generally persons are not sufficient to speak on such topics. To the next. Cheers

You made some decent factors there. I seemed on the web for the problem and found most people will go along with with your website.

very good publish, i actually love this website, carry on it

You made some respectable points there. I regarded on the web for the problem and found most individuals will associate with along with your website.

It's laborious to find knowledgeable individuals on this matter, but you sound like you understand what you're speaking about! Thanks

I'm often to running a blog and i really appreciate your content. The article has actually peaks my interest. I'm going to bookmark your web site and hold checking for brand new information.

I'm not easily impressed. . . but that's impressnig me! :)

o8I6CM AFAIC that's the best awnser so far!

Kudos! What a neat way of tihinnkg about it.

A million thnaks for posting this information.

I really liked your article. You should write more about that topic.

Really great article with very interesting information. You might want to follow up to this topic!?! 2011

Really interesting blog, keep up the good work!

Really interesting blog, keep up the good work!

Thanks for tris interesting article! I found it very interesting. You really make it seem so easy with your presentation.

I was very excited to see this site. I wanted to thanks for this great article! I definitely enjoying every little bit of it and I have you bookmarked to check out new stuff you post.

Thank you for this information! I used it for my diploma thesis =)

Thanks for the writeup about this. Greatly appreciated.

great job!!!!!!!!!!!!!!!!It is very useful and informative.well done guys

this is just out of the world, liked it a lot

you rocked

thanks a lot for this article. i learnt something new and different.

very good article

you can also use different made up shapes.if you cut one end of a square and stick that piece on the opposite end it will also make a tesselation.

yes that is true Mr Campbell my teacher taught me how to do it in grade 3. i still remember it

Just excellent want to read more

You might comment on the order system of the blog. You should chat it's splendid. Your blog audit would swell up your visitors. I was very pleased to find this site.I wanted to thank you for this

You might comment on the order system of the blog. You should chat it's splendid. Your blog audit would swell up your visitors. I was very pleased to find this site.I wanted to thank you for this

Good to become visiting your weblog again, it has been months for me. Nicely this article that i've been waited for so long. I will need this post to total my assignment in the college, and it has exa

Good to become visiting your weblog again, it has been months for me. Nicely this article that i've been waited for so long. I will need this post to total my assignment in the college, and it has exa

Great

amaz ing.... but you should elaborate mr campbell

I am really enjoying reading your well written articles. It looks like you spend a lot of effort and time on your blog. I have bookmarked it and I am looking forward to reading new articles. Keep up t

It is perfect time to make some plans for the future and it is time to be happy. I've read this post and if I could I desire to suggest you some interesting things or suggestions. Perhaps you could wr

It's late finding this act. At least, it's a thing to be familiar with that there are such events exist. I agree with your Blog and I will be back to inspect it more in the future so please keep up yo