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To be a Tesseract

Candidate Personal Code: gxf277

IB Mathematics SL

Date of Submission: 1/12/2018Table of Contents

Introduction ………………………………………………………………… 1

Discussion ………………………………………………………………….. 2

Part 1…………………………………………………………………. 2

Triangle………………………………………………………….2

Square……………………………………………………………8

Pentagon…………………………………………………………9

Part 2………………………………………………………………….. 10

Conclusion……………………………………………………………………11

Works Cited………………………………………………………………… 12

Introduction

The word tesseract intrigued me when I first watched movie “Avengers” in 2012. In the movie, creatures from space were trying to invade earth using tesseract’s power of opening a gate between space to space and brining their troops to earth. Avengers gathered by Tony Stark and Captain America is fighting against those troops and try to retrieve the tesseract to save the Earth. In the screen, The Tesseract was described as a glowing blue cube. At that time, I was curious how can that small cube have such a power to control the space and thought it is a fictional thing. When I saw this topic on the math subject list at 2016, I was astonished that tesseract is an actual geometric concept. I did pre-research on tesseract before I actually decide to do with it. When I searched about the tesseract, I found out that tesseract was already showed to the people in Iron man 2 with its structure diagram. The diagram describes that tesseract is not a single cube, but a cube inside the cube. Definitely, it was not a 3D model we used to know. It was a more complex geometric concept. This increased my interest in the topic. Thus, by using my math skills of sequence and geometry, I will investigate the identity of the tesseract and how it is created. Also, I will try to solve my curiosity of can other regular shapes that are not square can be a tesseract.

In this investigation, I will first analyze the vertices, edges, and faces of tesseract to have enough background knowledge about tesseract. Tesseract is a 4th dimension shape that its base is a square. A dimension refers to how many mutually perpendicular directions that an object can be measured (“Dimension”). A line is a 1st dimension because we can count only 1 perpendicular direction. A point itself is zero dimension because there is no direction for a single point. All polygons, solids, and shape went through point and line before they were formed, thus they share the same 1st and zero dimension. A shape like a square, triangle, and pentagon are the 2nd dimension. This is because they have length and width, which has two perpendicular directions of movement. As dimension increases more perpendicular directions of movement are available. Solid is the 3rd dimension because there is height, length, and width. The 4th dimension is a dimension that can count 4 perpendicular directions from an object. However, it is hard to observe 4th dimension accurately because our observable dimension is limited to height, length, and width. I used the data of vertex, edge, and face of 4th and nth dimensional solids predicted by mathematicians to investigate the pattern of terms. Given that, I will use other regular polygons, such as triangle and pentagon to research about whether their 4th dimension solids have similar components to tesseract.

Vertex, edge, and face are the main component of a solid. Vertex is a point where two or more lines or curve meet each other and create an angle. Edge is a line segment that is between two vertices. A face is a third-dimensional concept. The face is an area where length and width are perfectly enclosed. It is a surface area of a solid.

A sequence is another concept that is important in this exploration. Sequence refers to a list of terms in particular order. In math class, I learned two common sequences: arithmetic sequence and geometric sequence. An arithmetic sequence is a sequence that difference between the consecutive terms is equal. The formula of arithmetic sequence is an = a1+(n-1)d. “d” in the formula represents the common difference. “n” in the formula refers to the nth term of the sequence and “a1” refers to an initial number of the sequence. A geometric sequence is a sequence that has the common ratio between consecutive terms. The equation of geometric sequence is

an =a1·r(n-1). “r” in the formula represents the common ratio. “n” and “a1” has the same meaning to arithmetic sequence. Alternatively, a polynomial function can also work as determining the pattern too. More details about getting equation will be further explained in the discussion part.

Discussion:

The investigation will be held in two parts. In the first part, I will use triangle, square, pentagon as a base shape to create 4th dimension to see whether it meets the same components of tesseract. The pattern in terms will be determined by analyzing the change of vertices, edges, and faces. In the second part, I will compare the vertices, edges, and faces of regular triangle and regular pentagon’s 4th dimension shape with tesseract.

Part 1.

Triangle:

Below chart presents the number of vertices, edges, and faces from first to fourth dimension of triangle.

Change of vertex by increasing dimension

To find a pattern, I need to know whether there is a common difference or common ratio. When I get common difference the sequence will be an arithmetic sequence. On the other hand, If there is common ratio the sequence will be a geometric sequence. However, there is also a case that pattern is not arithmetic or geometric sequence.

Find the pattern

Common difference test (d): Common difference is a constant difference between previous term and current term in consecutive pairs.

d= a2-a1 d= a3-a2

= 3-2 = 4-3

= 1 = 1

Therefore, the common difference of the terms in vertex is 1.

Common ratio test (r): Common ratio is a constant ratio between previous term and current term in consecutive pairs.

r= a2 ÷ a1 r= a3 ÷ a2

= 3 ÷ 2 = 4 ÷ 3

= 1.5 = 1.33

The ratio between 2nd term and 1st term and the ratio between 3rd term and 2nd term is different. Therefore, there is no common ratio. This means the terms of vertex is not a geometric sequence.

2. Generalize into an equation

The equation for arithmetic sequence is an = a1+(n-1)d. In this case, “a1” represents vertex of 1st dimension, “n” represents dimension, and “d” represents common difference. The next step is to substitute the values from the table to the equation of arithmetic sequence.

a1 = 2

d = 1

So the equation is:

an = 2 +(n-1)1.

3. Verify the validity of the equation

The 5th dimension of triangle is 5-simplex. 5-simplex has 6 vertices (“5-simplex”). To validate my equation, I will test my equation with 5th dimension and compare with the real value.

The “n” value is 5, because it is a 5th term. Therefore, I substituted 5 to “n” and test the equation.

a5 = 2 +(5-1)1

= 6

Since the data got from the equation equals to real value, I can determine that arithmetic sequence is an exact pattern to describe the relationship between dimension and number of vertex.

Change of edge by increasing dimension

Common difference test (d):

d= a2-a1 d= a3-a2

= 3-1 = 6-3

= 2 = 3

There is no common difference.

Common ratio test (r):

r= a2 ÷ a1 r= a3 ÷ a2

= 3 ÷ 1 = 6 ÷ 3

= 3 = 2

There is no common ratio exist.

It can be concluded that there is either no pattern for the number of edges or there is another pattern that is not geometric or arithmetic sequence. For this case, the first thing to do is to input the data on the table into a scatter plot and figure out the best regression line.

Scatter plot:

*Scatter plot was created by using TI-Nspire™ (Graphing calculator)

The scatter plot presented on the calculator is hard to see that it is linear. Rather it is exponential than linear. In TI-Nspire™, a user can select which regression line user will use for the scatter plot. Because it is exponential, I used 3 regression line to test it: exponential function, quadratic function, and cubic function.

Exponential function:

In exponential regression line, the equation is y = 0.547723 • (2.13847)x.

Quadratic function:

In quadratic regression line the equation is y=0.5×2 +0.5x+0.

Cubic function:

In cubic regression line the equation is y= 0x3 +0.5×2 +0.5x+0.

After I get 3 possible regression line, the next step is to verify which equation has the closest value to actual value. 5th dimension of a triangle has 15 edges (“5-simplex”).

Let x represent the dimension. If we substitute 5 to the equation y = 0.547723 • (2.13847)x, the value of y is 24.495. The error range between actual and regression line is about 9.5. For quadratic equation, it was y=0.5(5)2 +0.5(5)+0. The result was 15. The equation of cubic function is same as quadratic equation because the multiplication of any x value put in to x3 and 0 is always 0, which means there is no value. Consequently, it can be determined that edges in triangle has quadratic function as their pattern.

Change of face by increasing dimension

Common difference test (d):

d= a2-a1 d= a3-a2

= 1-0 = 4-1

= 1 = 3

There is no common difference exist.

Common ratio test (r):

r= a2 ÷ a1 r= a3 ÷ a2

= 1 ÷ 0 = 4÷ 1

= undefined = 4

There is no common ratio exist.

Scatter plot:

The points in scatter plot is having a shape of exponential. So, exponential, quadratic, and cubic function are tested. However, exponential function is not suitable because the range of exponential function should always be bigger than 0.

Quadratic

In quadratic regression line the equation is y=1.25×2 – 2.95x+1.75.

Cubic

In cubic regression line the equation is y= 0.166667×3 – 2.8 E-12 x2 + 0.166667.

The 5th dimension of triangle has 20 faces (“5-simplex ). To find out which regression line fits better, I substituted 5 in to x variable for each regression line. When 5 is inputed in quadratic equation y=1.25(5)2 – 2.95(5)+1.75, the result was 18.25. On the other hand, when 5 is inputed in cubic equation y= 0.166667(5)3 – 2.8 E-12 (5)2 + 0.166667, the result was 20.667. The result from cubic equation is closer to actual value than the quadratic equation. Therefore, the pattern of faces is cubic function.

To summarize the process, geometric and arithmetic sequence was initially tested for the data. If there is no presence of them, using a graphing calculator to find the regression line to predict the pattern of vertices, edges, or faces of 5th-dimensional shape and compared with the actual data. This process was repeated for square and pentagon.

Square:

Change of vertex by increasing dimension

Change of edge by increasing dimension

Change of face by increasing dimension

Equation of vertex:

– Type of pattern: Geometric sequence

– Common ratio (r) = 2

– Equation: an =2·2(n-1)

Equation of edge:

– Type of pattern: Cubic function

– Regression Line: y= 1.16667×3 – 4.5×2 + 8.3333x – 4.

– Margin of error: 8.99

Equation of faces:

– Type of pattern: Cubic function

– Regression Line: y = 1.5×3 +7×2 +11.5x -6

– Margin of error: 16

Pentagon

Change of vertex by increasing dimension

Change of edge by increasing dimension

Change of face by increasing dimension

The 5th dimension of pentagon is unknown so it is impossible to test the validity of the equation.

Part 2

In the second investigation, I will compare the components of 4th dimension of each shape.

The value of vertex, edge, and face for each 4 dimension shape does not share any similar value with each other. Therefore, triangle and pentagon cannot be a tesseract even when they are made to 4th dimensional shape.

Conclusion:

In this exploration, my aim was to know about the tesseract itself and try another regular polygon as a base to see whether their 4th dimensional shape can have same components to tesseract. One of the exploration was to find about vertices, edges, and faces of the 4th dimensional shape of triangle and pentagon to compare with tesseract. However, other regular shapes didn’t have the same number of vertex, edge, and face with tesseract. Therefore, triangle and pentagon cannot become a tesseract. The second exploration was created to investigate the pattern of vertices, edges, and faces from the first dimension to 4th dimension of triangle, square, and pentagon. Using the data, I found out that pentagon and triangle have a totally different pattern of increase from line to its 4th dimensional solid. This once again strongly supports that triangle and pentagon cannot be a tesseract.

The strength of my exploration can be seen in discussion part. While I was doing my investigation, I was challenged to the case where there were no geometric or arithmetic sequence in the terms. However, I used regression line to determine whether the numbers have another pattern. In addition, I was able to find the best polynomial equation to predict the vertices, edges, faces of 5th-dimensional shape. For example, the edge of the triangle from line to 4th dimension had a pattern of quadratic. This was proved because the value of the 5th dimension and the predicted value are similar.

In contrast, there is also some weakness in my exploration. There is a limitation to my process of getting the equation from the regression line. First, I was not able to check the validity of the equation for the pentagon because the 5th dimension of the pentagon is unknown. Second, when I was finding a pattern of faces on the 4th dimension of the triangle, the cubic equation was selected to be the best regression line. The problem occurs here because I used the graphing calculator and graphing calculator depicts the only certain area of the function. Therefore, it is not sure that data will be matched to the real value when the dimension is increased more than 5. In addition, if I have more time for this exploration, I would try my equation with higher dimensions over than 5 and test the validity of my equation precisely.

Works Cited:

“Five-Dimensional space.” Wikipedia, Wikimedia Foundation, 27 Aug. 2017, en.wikipedia.org/wiki/Five-dimensional_space#Five-dimensional_geometry. Accessed 10 Sept. 2017.

Stapel, Elizabeth. “Arithmetic & Geometric Sequences.” Arithmetic & Geometric Sequences | Purplemath, www.purplemath.com/modules/series3.htm. Accessed 10 Sept. 2017.

“What is a four dimensional space like?” Four dimensions, www.pitt.edu/~jdnorton/ teaching/HPS_0410/chapters/four_dimensions/index.html. Accessed 10 Sept. 2017.

“5-Cell.” Wikipedia, Wikimedia Foundation, 1 Sept. 2017, en.wikipedia.org/wiki/5-cell. Accessed 10 Sept. 2017.

“The First Five Dimensions.” TheScienceClassroom,

thescienceclassroom.wikispaces.com/The First Five Dimensions.

“Sequence.” Definition of, www.mathsisfun.com/definitions/sequence.html.

“5-Simplex.” Wikipedia, Wikimedia Foundation, 26 Dec. 2017, en.wikipedia.org/wiki/5-simplex.

“Dimension.” Dimension | Think Math!, thinkmath.edc.org/resources/glossary/dimension.

“120-Cell.” From Wolfram MathWorld, mathworld.wolfram.com/120-Cell.html.

“Pentatope.” From Wolfram MathWorld, mathworld.wolfram.com/Pentatope.html.

“Tesseract.” From Wolfram MathWorld, mathworld.wolfram.com/Tesseract.html.

Vertices, Edges and Faces, www.mathsisfun.com/geometry/vertices-faces-edges.html.