Pre-Calculus 0809

The method in which i created this recursion was with a computer program known as Inventor. In this program I created a “puzzle cube” or a group of components comprised of cubes that when put together, the components form into a three by three cube. To simulate this recursion, I used the the finished puzzle cube as the base cubes to create the components for a larger three by three cube. Then this cube is used to make the components for the next cube.

This process was used to create three cycles of this process. each step should theoretically look exactly like the step before it excluding the size of the cube. The Inventor program provided some difficulty presenting the smaller pieces that make up the cube. The computer uses the theoretical idea and shows the components as a singular piece instead of the individual pieces that it is comprised of. The equation for the number of cubes in each step is AN= (3n)^3.

To make my recursions project my first step was to find a good picture. After  found a good picture i knew my teacher would like i need to ues it in recursion some how. to do this i printed off a copy of the picture.

After thast i found a camera and took a picture of me holding the picture. I then downloaded the picture on to the computer and printed a picture of that one. After this i printed it agine. Then once again i took a  picture of me holding the picture, downloaded it, and then printed again. The final step was to tape them on to a sheet of paper so they would stay together and i was done.

For my recursion I kept it very simply just basic rectangles inside of each other.  I didn’t have a picture of me holding a sign and I am not that artistic. It was just a simply rectangle in a rectangle over and over again.

In most examples of recursion there is more to it pictures being repeat like i said but it is really hard to grasp the concept of this without a picture.  Thats why even though mine is a basic rectangle repeated it still shows what recursion can be in the simpliest form.

For my recursion project, i used an old picture from a volleyball game this past fall. I am standing in the picture with my arms in the air. When i shrunk each picture by about 1/9th of the picture before it, it looked like i was holding the picture in my hands.  I thought it was a pretty clever idea and it turned out better than i had expected.

However, recursion itself is a concept that is hard to understand. Recursion is defined as a function that is defined by itself. This basically means that an object is repeated over and over again. With recursion defined as an object repeated over and over again, my project is an excellent example of it.

For my recursion project I decided to draw a sketch.  In the sketch a person is holding a frame in front of their face, and in the frame is a picture of a person holding a frame with a person holding a frame etc. I had originally decided to use a picture of my dog and make it recursive, but at the last minute I decided not to.

Recursion is when a subject is repeated in itself.  My sketch shows recursion because my picture is repeating within another picture.

okay about two weeks we were assigned a recursion project. What recursion is, is when a a subject is defined in the same subject. So i decided that i would do my project on taking pictures of my little sister three times over but my printer was not working for me. Thats when i decided to do my project one my aunts dog. By making her dog do the same pose over and over again then putting the picture in the same spot.

The reason that me example is recursive is because i have the same picture in it. Also the pictures are getting smaller of charlie with each new picture i take. The way that it is recursive also is because he is in the same spot with the same pose as the first picture. Recursion can be explained also by fractions, well fractions can also help explain how recusion works.

             For my recursion project I had no idea what I was going to do. It took a long period of deep thought to find out what I decided to do. I was looking down at my phone playing with the pictures. I was in my basement at the time watching Dane Cook, who is the funniest comedian on Earth. I thought of what recursion was, and there I had found it, my idea.

             I decided to draw my phone taking a picture of the same phone which took a picture of it again and etc. After I finished it looked really cool. It showed recursion by having a phone inside a phone and kept going. I had to balance the sizes of course so it took a little bit of measurement. I had to have the same ratio of the phone’s size but also had to make it smaller. Even though it seemed easy it did take a bit of work and was quite difficult. It was a fun project however and I’m glad that I did it.

My project shows recursion because it is a bunch of squares inside each other. I made this using paint. First I made a big square then I used the line making tool and made a line from the middle of the one side to the next. I did that for all four sides of the square. That would make a square rotated a little bit and I did that for all of it. So it shows recursion by a square inside a square but tilted a little bit everytime.

When this project was assigned i had the idea of taking a picture of myself 3 times in the same pose holding the picture (the classic recursion example). When I tried to start making this project I was having some trouble with my digital camera so I had to do something else instead. So for my project I drew an iPod inside the screen of an iPod and an iPod in the screen of another ipod. It actually turned out to look pretty cool.

This is an example of recursion because its a process of repeating objects in a self similar way. Each time you repeat the object in the other object it gets smaller. The Fibonacci Sequence is another great example of recursion. Also factorials are pretty recursive. Recursion rules!

For my recursion project i did a video! i took a couple videos of me in front of a tv. Then i used another camera to take a video of me sitting with the first video playing on the tv behind me. I also found a few pictures that show recursion. The last part of my video is my hand. I put my hand in front of the camera and hooked the camera up to the tv and i was filming what the tv was showing over and over again.

In my video you can see math because the video showed pictures that are the same but because of the depth the ones towards the middle look smaller and smaller. With math you could find out how much smaller the picture looks. Fractions are related to recursion, the smaller pictures are fractions of the starting picture.

The recursion project that i did, was pretty easy to do. First you must know what recursion is. Recursion is something that is defined by itself, ( so it is like a picture inside a picture) The way i did my project was by taking a picture and then printing that picture off. Then i took that picture and put it with the object i took a picture of to begin with, and i took a picture of that. I repeated that process 3 times.

This shows recursion by the pic i took being taken with the same object and it will be a picture inside a picture. This can be shown mathematically by using  fractions. This is possible because the first picture taken is one full unit. (like 9/9) the second picture is then taken with the first picture. the first picture will then be a fraction of the second picture. (like 1/9) The third picture is then taken with the second picture. the second pictue that is inside the third picture that is taken is also the same fraction as the first picture inside the second picture. (1/9)

When I heard about this recursion project i knew that i would have to take pictures to show recursion because i am not all that great with technology and I am even worse at art. Also, the picture project is a fun way to show creativity and show recursion at the same time.

To show recursion i took a picture of myself and then took a picture of myself holding that picture and repeated the process again. This is an example of recursion because it is smaller and smaller but is showing the same picture.

For starters, recursion isn’t exactly the easiest topic to learn, so when we got this project, I got a bit nervous. Because I’m not exactly the most creative person in the world (although I do talk incredibly fast and I always have a lot on my mind), I decided to go with the picture, in the picture, in the picture for my project. This shows recursion because the pictures get smaller and smaller, yet they remain the same image. After I got all three of my pictures developed, I put them on really pretty scrapbooking paper in the order they went in.

This is an example of recursion because recursion is a term used to describe a process of repeating objects in a similar way. For instance, in the picture above, it shows recursion because the man is holding a picture which contains another smaller picture of the picture, and an even smaller picture of the picture, and so on and so forth. This is a very classic example of recursion and as much as I like it, I only wish the background was pink

I got the idea for my recursion project from the Spaceballs excerpt and the multiple examples involving the holding of pictures. Recursion is a function that is defined by itself. This just means that an object is repeated over and over.

Instead of looking at a television or holding a picture of myself, I used a laptop. I used a laptop because it was small enough so that I could hold it. My picture was taken several different times in different poses, until I got one where I did not look ridiculously stupid. Then I took that picture and inserted it onto the blank screen of the laptop, the blank screen in that laptop, and so on. It repeats itself, therefore, it is recursion.

For my Recursion project it took me a while to think of  something to do.  Finally i thought of something simple,but i never thought I would be drawing because i am artistically challenged (I haven’t taken an art class since my eight grade year).  I decided to draw a tree because it is a real life example of recursion as a tree grows it continually branches off.  It doesn’t work as perfectly in nature as it did in my drawing.

It begins with a single trunk and then it splits into two and those parts split into two and so on and so forth.  As you keep going the branches get smaller and smaller and the tree gets wider and wider.  I kept drawing branches until they started to run into each other, but the tree could get expotentially taller and wider if there weren’t branches running into each other.

When I got the recursion project, I knew I would have to draw my example.  (I’m technologically challenged.)  I also knew that I wanted to do something dealing with a picture inside a picture.  I tried drawing people and scenes, but the details got complicated.  I suddenly thought of a picture in a textbook, words and lines (pretty simple).  A little while later, my example was born.  It is similar to the one below except for a picture of a picture of a……..textbook.

It shows recursion because the pictures get exponentially smaller and (could) go on forever.  It is also the same thing repeating within itself.  My example deals with math in a few ways.  I used a textbook (math i.e.) for my picture.  Also, the pictures get around two times smaller each time.

Recursion is an interesting topic to understand. It can get a little confusing since the function is defined by itself. The most common example is a picture within a picture. This is the real life example that I used. I started with a black-and-white picture of my hands. After I printed it out, I took another picture of me holding the first one. I repeated this process a couple of times. The last picture looks like this:

After all of the pictures were printed out, I mounted them in order on black paper to make them more presentable.

And in honor of Right to Read Week, check out Flotsam by David Wiesner.  This picture book is the story about a little boy who finds a camera washed up on the beach. He develops the pictures and the last one is a recursive shot of everyone who found the camera before him. Although it is meant for children, it is still a good example of recursion in real life.

My example of recursion is fairly simple…I started brainstorming and came up with the idea of doing exactly what recursions is defined as, something inside of itself. The word recusion created out of recursion. As i started i put on some music i heard the song All You Need Is Love by The Beatles and it inspired me to create the word love out of love.

The first step was to draw an outline of the word love. The next step was to simply use the line and write the word love over and over again to create a bigger version of the word.  This basically defines the word recursion. The mathmatical properties of this are just basically a repeated pattern.

For my Recusion Project inspiration i looked upon some websites and saw some really cool pictures of recursion happening. But i knew i could’nt do that cool of stuff so i went with a basic simplified version. I basically took a picture of my face and printed it out. Then i held that picture to my face again and took a picture and printed that out. I did this about 5 times till i got a spiral looking affect of faces.

This shows an example of recursion in its simplest terms byrepeating an object over and over again. I tried to show the best example from me by taking the face of muah, and repeating it till i got a spiral recursion.

This is Hayley Williams of Paramore. ♥
So, Paramore + music theory= recursion project inspiration. 

I discovered Paramore about 2 years ago.  Since then, they have become so awesome that even string quartets are covering their songs (yay, chamber groups!!!!).  Their song “Pressure” has been covered by the Vitamin String Quartet and redone in an acoustic version by the band.   I was originally going to play all three versions on top of eachother, but that sounded gross.  Then I tried to break them apart and introduce the next track after each time a chorus ended.  Well, that didn’t sound very good either.  It doesn’t exhibit the idea of recursion very well if you can’t even hear what the heck is going on.  So, I created a playlist with the songs in this order: Quartet version, acoustic version, original version.  This way, the songs build (in terms of music theory) as the playlist goes on just like those equations we did in class.  

The Quartet’s version consists of a basic bass line, harmony, and one line of melody.  It is pretty much the song in its simplest form.  The acoustic version takes the harmony and bass and puts it into chords played by two guitars.  The guitars also act as the rhythym section in this song (which the previous one did not have).  The melody sung with words rather than just played.  The original version consists of vocals with harmony, two guitars with distortion, a bass, and a drum section.  It also incorporates some little guitar licks not included in the acoustic version.  The rhythym and bass are much more complex in this song than either of the previous versions.   Each song adds new elements to the basic frame of the song and makes the new version even bigger and better than the last one.  You cannot get the original version without breaking it all down and using the basic music theory from the Quartet version.  As for the math behind all this…  This is music, and of course you can’t have music without math.  The biggest thing that changed in these songs every time was harmony.  Harmony is not just throwing notes together; only notes at particular intervals sound good together.  Also chords in a song must all be in the same key.  The key of a song is determined by the notes being used and the intervals of whole and half steps between them.  Time signature is another HUGE part of music, although it usually only consists of counting to 4 or 6.  Without it music would always fall apart.