Spiralateral P.O.W.
Problem Statement:
A spirolateral is a figure that is created through the repetition of a set amount of steps in a certain direction, whether the direction is clockwise to counterclockwise or in terms of the degree is turns by. There are several different variables that could change the product, the final figure, of the spirolateral. This applies to my question, or the variable that I investigated, which was how the different degrees that the spirolateral is turned by affects the shape of the spirolateral.
Process Description:
To try and test this, I had three different pages with the same sequence of numbers on each. With this, each degree would show how the spirolateral would differ with different degrees used.
Results & Conclusions:
In the diagrams with the sequence of 1, 2, 3, 4, one can clearly see the difference that the degrees do to the spirolateral. In the 90 degree spirolateral, the spirolateral is unending and has the shape of a quadrilateral throughout. For the 60 degree spirolateral, it rotates more than any of the others and is made of triangles. In the 45 degree spirolateral, it gives off the image of a polygon with a small line on the inside of it. There seems to be a constant pattern for the 60 degree and the 90 degree spirolaterals, the 60 degree seems to be more circular in a way, with curves on some of them. The 90 degree is more ‘right-angled’ with squares and rectangles being the main visual in the spirolateral. The 45 degree spirolateral, however, seems to be more like a polygon, a broad description for a broad range of spirolaterals in this degree. With rectangles, diamonds, and curves in some of the spirolaterals, thus making it difficult to assign a certain pattern to it. I concluded this from the first six number sequences that I made for each of the degree. The 60 degree one may seem a bit off due to the difficulty creating the spirolaterals, to make this easier I had to look for a program that would assist me in creating the main idea for the shape of the spirolateral. But my main reasoning for the conclusions I made would be due to the shape and images that I see in the spirolaterals. The conclusions I made, are the same, if not similar, to the rules for the spirolaterals. I also realized halfway through that the degrees that I used were only acute angles, except for the right angle, and I questioned how the sequence would differentiate with obtuse angles.
90 Degree-
Self-Assessment:
The two Habits of a Mathematician that I used would be to solve a simpler problem and to be systematic. I solved a simpler problem because I used simple numbers in the function such as one, two, three, etc. instead of thirty, seventy-five, etc. I also started off with the sequence staying the same throughout and adding the next number in the sequence to be one more than the previous. This is similar as to how I am systematic because I was testing the different degrees of a spirolateral and the different degrees that I used compared to a similar sequence on another degree shows this. This is why I think I deserve a 10/10.
A spirolateral is a figure that is created through the repetition of a set amount of steps in a certain direction, whether the direction is clockwise to counterclockwise or in terms of the degree is turns by. There are several different variables that could change the product, the final figure, of the spirolateral. This applies to my question, or the variable that I investigated, which was how the different degrees that the spirolateral is turned by affects the shape of the spirolateral.
Process Description:
To try and test this, I had three different pages with the same sequence of numbers on each. With this, each degree would show how the spirolateral would differ with different degrees used.
Results & Conclusions:
In the diagrams with the sequence of 1, 2, 3, 4, one can clearly see the difference that the degrees do to the spirolateral. In the 90 degree spirolateral, the spirolateral is unending and has the shape of a quadrilateral throughout. For the 60 degree spirolateral, it rotates more than any of the others and is made of triangles. In the 45 degree spirolateral, it gives off the image of a polygon with a small line on the inside of it. There seems to be a constant pattern for the 60 degree and the 90 degree spirolaterals, the 60 degree seems to be more circular in a way, with curves on some of them. The 90 degree is more ‘right-angled’ with squares and rectangles being the main visual in the spirolateral. The 45 degree spirolateral, however, seems to be more like a polygon, a broad description for a broad range of spirolaterals in this degree. With rectangles, diamonds, and curves in some of the spirolaterals, thus making it difficult to assign a certain pattern to it. I concluded this from the first six number sequences that I made for each of the degree. The 60 degree one may seem a bit off due to the difficulty creating the spirolaterals, to make this easier I had to look for a program that would assist me in creating the main idea for the shape of the spirolateral. But my main reasoning for the conclusions I made would be due to the shape and images that I see in the spirolaterals. The conclusions I made, are the same, if not similar, to the rules for the spirolaterals. I also realized halfway through that the degrees that I used were only acute angles, except for the right angle, and I questioned how the sequence would differentiate with obtuse angles.
90 Degree-
- Quadrilateral Shape
- Is Never-Ending for Factors of Four
- Consists of Rectangles and Squares
- Circular
- Rotates the Shape until it Ends
- Always Ends
- All Polygons
- Inconsistent Shape
- Always Ends
Self-Assessment:
The two Habits of a Mathematician that I used would be to solve a simpler problem and to be systematic. I solved a simpler problem because I used simple numbers in the function such as one, two, three, etc. instead of thirty, seventy-five, etc. I also started off with the sequence staying the same throughout and adding the next number in the sequence to be one more than the previous. This is similar as to how I am systematic because I was testing the different degrees of a spirolateral and the different degrees that I used compared to a similar sequence on another degree shows this. This is why I think I deserve a 10/10.