Spiral-aterals
Problem Statement:
This problem is about Spiral-aterals. This is a sequence of segments or lines that form a spiral-like shape. Each one is based on a sequence of numbers. One example is 3,2,4. To draw the spiral, you need graph paper. The starting direction is always ‘up’ on the paper. You would first go ‘up’ three spaces on the graph paper. Then turn 90 degrees clockwise (right) and go two spaces. Next, you would turn another 90 degrees right and draw another line four degrees down from the end of the second line. Now you continue this pattern, turning 90 degrees right each time, until you end where you started.
Our assignment for this problem was to explore the ideas and patterns of these spirals. Some things our teacher suggested for us to look at were try different sequence lengths, make up new rules, and to look for patterns.
I found this problem fairly fun to explore because it was easy and I just got to make patterns and draw on graph paper. The first thing that I did was just making spirals. I did lots of different sequences, such as 2,4,6 and 1,2,3,4,5. The next thing that I did was instead of going clockwise, I chose to go counter-clockwise. Then, I experimented by going one full sequence clockwise, the next counter, and so on. Below is the spiral of 1,2,3,4,5 and the counter/clockwise switch ‘spiral’/stairs of the 1,2,3,4,5 sequence.
Then, all of my classmates were using whole phone numbers as their sequences. I got the idea from a friend for a creative way to use zeros in a sequence. You would start you sequence however it starts, and when the zero comes in you act like it is a number. You turn and act like you are going to make a line, but since it is a zero, you just turn the paper again and go on to the next number. I think of it like a 180 degree turn instead of the usual 90 degrees. Then I started looking at how putting a zero in a sequence affected the spiral. I used the sequence 1,2,3,4 and then put zeros in it and saw what happened. Below is the regular sequence of 1,2,3,4 and the sequence of 0,1,2,3,4.
The spiral going down becomes a spiral that turn in on itself when you put the zero in. Next are the sequences 1,0,2,3,4 and 1,2,0,3,4 and 1,2,3,0,4.
Since there is no pattern that really happens here, I don’t really have a solution. It goes from a spiral to coming in on itself, to getting more spread out, to more condensed, to a tiny little spiral. I can’t find anything here to compare to so the only solution that I have is that making these spirals was really fun and I enjoyed drawing on graph paper.
For this problem, I think that I deserve a 10/10. This is because I actually tried to dig deeper and answer a wondering that I had. I experimented a lot with different numbers. I know that I do not have a solution to this problem, but I doubt that many people do since this is a pretty open problem and people did all sorts of things.
The rules that I used were just the rules that were given to me, although I made up my own way to experiment with the zeros. Some questions that occurred to me was if it was possible that the zeros don’t do anything, or if it works better when going counter-clockwise. I was not able to experiment these within the time I had, so I don’t know the answers to these questions.
The Habits of a Mathematician that I used when solving this problem were solve a simpler problem, look for patterns, and visualize. I used solve a simpler problem because I had a wondering that was pretty simple to explain and look at, after all this problem is all about spirals in graph paper. I also used look for patterns because that is exactly what I was doing to get to a solution. As well, I also used visualize because I drew a lot because we were looking at all of these patterns on paper.
This problem is about Spiral-aterals. This is a sequence of segments or lines that form a spiral-like shape. Each one is based on a sequence of numbers. One example is 3,2,4. To draw the spiral, you need graph paper. The starting direction is always ‘up’ on the paper. You would first go ‘up’ three spaces on the graph paper. Then turn 90 degrees clockwise (right) and go two spaces. Next, you would turn another 90 degrees right and draw another line four degrees down from the end of the second line. Now you continue this pattern, turning 90 degrees right each time, until you end where you started.
Our assignment for this problem was to explore the ideas and patterns of these spirals. Some things our teacher suggested for us to look at were try different sequence lengths, make up new rules, and to look for patterns.
I found this problem fairly fun to explore because it was easy and I just got to make patterns and draw on graph paper. The first thing that I did was just making spirals. I did lots of different sequences, such as 2,4,6 and 1,2,3,4,5. The next thing that I did was instead of going clockwise, I chose to go counter-clockwise. Then, I experimented by going one full sequence clockwise, the next counter, and so on. Below is the spiral of 1,2,3,4,5 and the counter/clockwise switch ‘spiral’/stairs of the 1,2,3,4,5 sequence.
Then, all of my classmates were using whole phone numbers as their sequences. I got the idea from a friend for a creative way to use zeros in a sequence. You would start you sequence however it starts, and when the zero comes in you act like it is a number. You turn and act like you are going to make a line, but since it is a zero, you just turn the paper again and go on to the next number. I think of it like a 180 degree turn instead of the usual 90 degrees. Then I started looking at how putting a zero in a sequence affected the spiral. I used the sequence 1,2,3,4 and then put zeros in it and saw what happened. Below is the regular sequence of 1,2,3,4 and the sequence of 0,1,2,3,4.
The spiral going down becomes a spiral that turn in on itself when you put the zero in. Next are the sequences 1,0,2,3,4 and 1,2,0,3,4 and 1,2,3,0,4.
Since there is no pattern that really happens here, I don’t really have a solution. It goes from a spiral to coming in on itself, to getting more spread out, to more condensed, to a tiny little spiral. I can’t find anything here to compare to so the only solution that I have is that making these spirals was really fun and I enjoyed drawing on graph paper.
For this problem, I think that I deserve a 10/10. This is because I actually tried to dig deeper and answer a wondering that I had. I experimented a lot with different numbers. I know that I do not have a solution to this problem, but I doubt that many people do since this is a pretty open problem and people did all sorts of things.
The rules that I used were just the rules that were given to me, although I made up my own way to experiment with the zeros. Some questions that occurred to me was if it was possible that the zeros don’t do anything, or if it works better when going counter-clockwise. I was not able to experiment these within the time I had, so I don’t know the answers to these questions.
The Habits of a Mathematician that I used when solving this problem were solve a simpler problem, look for patterns, and visualize. I used solve a simpler problem because I had a wondering that was pretty simple to explain and look at, after all this problem is all about spirals in graph paper. I also used look for patterns because that is exactly what I was doing to get to a solution. As well, I also used visualize because I drew a lot because we were looking at all of these patterns on paper.