Exploring the Fourth Dimension
Lori Alvin
‘Actually, what worries me is something quite different,’ said Vikki.
‘Does this
multidimensional stuff let you do anything useful? Or is it just intellectual fun and games?’
(Flatterland, Stewart)
Have you ever wondered what the fourth dimension is, or what it looks like? Do you think that
time travel might have something to do with the fourth dimension? Then Exploring the Fourth
Dimension is the right course for you. Throughout the nine weeks we will explore the worlds of
lower dimensional “creatures” and you will develop your own idea of what the fourth dimension
is. This class will force you to think outside the box (the three-dimensional kind), and explore
math from a different perspective. Students will use journals and on-line discussions to analyze
and defend their ideas of the fourth dimension. Students will also investigate other areas of math
ranging from geometry and fractals to the shape of the universe and paradoxes. Students will
piece together all this exciting, new knowledge into a magazine, which will serve as the final
project.
Weekly Breakdown of Topics
Weeks 1-4
Looking down into the 2nd dimension
 Weeks 1-2 First Impressions
 Weeks 3-4 A 2-D World?
Weeks 5-6
The Fourth Dimension (and time travel, etc.)
 Week 5 What is the Fourth Dimension?
 Week 6 Time Travel
Weeks 7-9
Higher Dimensions and Other Math
 Week 7 Expanding Horizons
 Week 8 Changing Perceptions
 Week 9 Sharing the knowledge
End Goal: Create a magazine about multiple dimensions with other students in the course.
Reading selections:
Flatland (classic/medium difficulty) 2 weeks—online for free
Planiverse (novel/ easy to medium difficulty) 2 weeks
Portions of The Fourth Dimension (easy to read)—provided
And He Built A Crooked House (short story/easy)—provided
Flatterland (novel/ easy to read but difficult to understand) 3 weeks
Other various articles, excerpts from books, and short stories—provided
One of the most important things in mathematics is sharing your ideas. One major way
mathematicians share their ideas is by writing them down. For this reason, there is a lot of
writing and sharing of your ideas with others. Many people do not realize how much writing
mathematicians do, but it is an excellent way to express any ideas you may have. Also,
remember that no idea is too crazy to share; there have been many mathematical and scientific
ideas that we embrace as fact today that were initially thought to be false.
Week – to – Week Schedule
Week 1
Weekly Theme: First Impressions
Reading: Read Part I of Flatland by Abbott, which can be found online at:
http://www.ibiblio.org/eldritch/eaa/FL.HTM or
http://nedwww.ipac.caltech.edu/level5/Abbott/Abbott_contents.html
Journal: What is a dimension? What do you think the fourth dimension is? Why is it important
to study dimensions? (I am not grading you for accuracy, and I do not want you to give me a
dictionary definition. I am looking for your first impressions and ideas; you can also mention
what those around you think.) Explain your answer in a well-formed paragraph, giving examples
and reasons for your opinions.
Thought Provoker: Describe what you think a “creature” from the fourth dimension would look
like. How would he act, and what would he do every day? Describe what you think would
happen if this “creature” visited Earth. You should write this from the creature’s point of view.
(This should be at least 2-3 well-formed paragraphs.)
Project: Introduce yourself to your assigned group. Be sure to mention your school, hobbies,
favorites, etc. (anything to help them get to know you). Then introduce your “creature” to the
group. Share what it is like in a day in the life of your “creature” and what happened when he
tried to visit Earth.
Choose Two Tasks from the list below:
Task 1. (Opinion-based Writing) Flatland was written not only as a mathematical book, but also
as a social satire…how does it make you feel how women and lower classes were portrayed?
Compare this to today’s society.
Task 2. (Creative Writing) What questions do you have about the Flatland world that were not
answered by the text? Explain, with your current knowledge of Flatland, how you think these
questions would be answered by A. Square, if you asked him.
Task 3. (Creative Writing) Describe our world to A. Square. Remember, he is 2-D and may not
understand, so be thorough.
Week 2
Weekly Theme: First Impressions
Reading: Read Part II of Flatland by Abbott, which can be found online at:
http://www.ibiblio.org/eldritch/eaa/FL.HTM or
http://nedwww.ipac.caltech.edu/level5/Abbott/Abbott_contents.html
Journal: In Flatland, A.Square visits Lineland and tries to explain to them about his higher
dimension. Rewrite the chapters about Lineland from an inhabitant’s point of view.
Thought Provoker: Generate a list of at least 5 thorough questions that you would ask your
creature from last week, if you were able to meet him. For each question, you should generate
an answer you think that the creature would give you. Each of your creature’s answers should be
at least one well-formed paragraph (if it is not possible to answer in at least one paragraph, your
question was probably not thorough enough).
Project: Describe what a day in your life would be like if you were restricted to 2 physical
dimensions. Go through your entire day and explain how anything would change in each
activity. Otherwise, explain why the activity would remain the same. Remember, almost
everything should go down one dimension: if you are playing checkers, for example, you would
not have a board that is length by width, it would just be length. Be as thorough as you can.
Share your day in 2-dimensions with the members of your group.
Choose Two Tasks from the list below:
Task 1. (Reflective) Write about a time that you tried to share your new-found knowledge in an
area with friends or family and they just did not believe you or thought you were crazy.
Task 2. (Creative Writing) Many questions for you may have been unanswered in this story.
Write an additional chapter that summarizes or brings closure to your questions.
Task 3. (Making Connections) Think about what the sphere has to say about the possibility of a
fourth dimension. Compare this to what people first thought about the idea that the earth
revolved around the sun, and not the other way around. Who is the author trying to compare the
sphere to?
Week 3
Weekly Theme: A 2-D World
Reading: Read chapters 1-6 of The Planiverse by Dewdney.
Journal: Compare and contrast the differences between the Planiverse inhabitants and the
Flatland inhabitants. Which world do you think is “more advanced”? Defend your answer with
quotes and examples from both texts.
Thought-provoker: What are some items that we take for granted in our 3-D world that you
would miss in the 2-D world? Can you describe/create an object that would serve the same
purpose in the 2-D world? If possible, can you describe these same objects in a 4-D world?
Project: Share your 2-D and 4-D items from the thought-provoker with your group. Here, do not
be afraid to question each others’ understandings of their own items; you may have a question
about the item that they did not explain when describing it. Try to expand on the ideas presented
by your group to ensure you have a full understanding of how different our common items may
be in a 2-D and 4-D world.
Choose One Task from the list below:
Task 1: (Reflective) Write about what you hope to learn about higher dimensions that can be
explained by looking at lower dimensions. How can The Planiverse help you in this task? You
may include anything you have read so far in the book or mention any questions you still have
after reading Flatland.
Task 2: (Connecting and Analyzing) On the Planiverse, there are many obscure names of people
and places. Determine the significance of these names and explain what extra aspect or insight
these names bring to the book.
Task 3: (Connecting and Analyzing) Why do you think that the Ardeans speak in such a distinct
manner? That is, why do they say “Those my parents, grandmother and sister are” instead of
“Those are my parents, grandmother and sister” as on page 11? Think about the time period this
book was written in, and also explore the pros and cons of why an author may choose to use such
dialect.
Week 4
Weekly Theme: A 2-D World
Reading: Finish The Planiverse by Dewdney.
Journal: Write a 5-paragraph essay (see format) comparing and contrasting Flatland and The
Planiverse. Your paper should include at least 2 quotes from EACH book.
Thought-provoker: Pretend you are a New York Times Book Reviewer and write your own
review for the book. This should include a brief overview of the book, and focus mainly on your
opinion. It should be at least 3 paragraphs long, and include any lessons that you think the
author was trying to convey.
Project: Share your book review with your group.
Choose One Task from the list below:
Task 1: (Understanding and Connecting) Answer all the attached questions about The Planiverse
(file to be attached here).
Task 2: (Understanding and Connecting) Draw a map of the Planiverse world as you have
interpreted it through the descriptions and pictures within the novel. You must be able to scan
this picture into a computer for submission to choose this task.
Task 3: (Reflective) What did you learn from The Planiverse about higher dimensions that you
were not able to learn from Flatland? Use quotes and examples from both texts to defend your
opinion.
Week 5
Weekly Theme: What is the Fourth Dimension?
Reading: Read each of the articles I have posted for you.
Journal: Provide thorough answers for each of the questions from “The Fourth Dimension”
article. By thorough, I mean that, unless otherwise stated, you should answer in full sentences
and cite examples from the resources we have used so far.
Thought-provoker: Create an encyclopedia entry for “the fourth dimension”. This should
include definitions, examples, and anything else that is generally included in an encyclopedia.
Project: Compare all your encyclopedia entries and as a group come up with your best
encyclopedia definition.
Choose One Task from the list below:
Task 1: Write a letter to your best friend summarizing what you have learned about the fourth
dimension this week. Actually give this letter to your friend and ask them to write you back in
response. You should submit (in a typed format) your letter and either include or summarize
your friend’s reaction.
Task 2: (Creative) After reading “And He Built a Crooked House…”, try to create your own
short story about a fourth dimensional house.
Task 3: (Analysis) It is extremely hard to visualize a fourth spatial dimension since we live in
three dimensional world. Looking back over everything you have read thus far in the course,
explain how the use of pictures has aided (or possibly hindered) your understanding. Has there
been any specific graphic that has caught your attention? Most of the book/article by Banchoff is
comprised of computer graphics and other images. How do these images aid in your
understanding of higher dimensions?
Week 6
Weekly Theme: Time Travel & other 4th Dimension Oddities
Reading: Read each of the articles I have posted for you.
Journal: Write a one page typed reflection about how time and space travel is portrayed in
movies and other books. Does this conflict with what you have read and debated this week?
Why do you think that humans are so interested in time travel?
Thought-provoker: Write your own unique time travel paradox.
Project: Share your time travel paradox with your group.
Choose One Task from the list below:
Task 1: (Mathematical Understanding) Try to explain, in a few well-constructed paragraphs,
how you could construct a hypercube if the fourth dimension is considered to be time (this is
tricky, so take your time and try to be thorough).
Task 2: (Analysis) Compare and contrast the following short stories that you were to read this
week: “The Appendix and the Spectacles”, “Star, Bright”, and “The Vanishing Man”. In doing
this, you are encouraged to look up the time periods in which these were written and any
information about the authors that you think may help in your analysis. One idea is to also
compare these stories with science fiction that has been written more recently.
Task 3: (Creative) Write a poem about the fourth dimension and time travel.
Week 7
Weekly Theme: Expanding Horizons
Reading: Read the preface and chapters 1-6 of Flatterland by Stewart.
Journal: ‘Actually, what worries me is something quite different,’ said Vikki. ‘Does this
multidimensional stuff let you do anything useful? Or is it just intellectual fun and games?’
(Flatterland, Stewart) Answer this question for Vikki. You should provide at least 3 quotations
from books and articles from this course, or you may use outside resources if you need/want to.
Project: Your final assignment for this course will be to create a magazine about dimensions
and other related mathematical topics from this course. This magazine will be made with your
group that you have been assigned to throughout this course. You are responsible for dividing
your work amongst yourselves; my advice to you is to stay in constant contact with each other
through the Moodle Village so that you are able to have a phenomenal product in the end.
Choose One Task from the list below:
Task 1: (Reflective) Write 3 paragraphs describing the importance of having a female
protagonist.
Task 2: (Comparing and Contrasting) Compare and contrast our society with the Flatland
society, which basically values those with more sides. How is America similar to this, and how
are we able to transcend beyond this (if we are able to)?
Task 3: (Reflective) In “A Hundred and One Dimensions”, is the People’s lack of understanding
about higher dimensions similar to any thoughts you have ever had about higher dimensions? If
not yourself, someone you know? Explain.
Week 8
Weekly Theme: Changing Perceptions
Reading: Read chapters 7-12 of Flatterland by Stewart.
Journal: Choose one topic that was introduced in Flatterland and research it further. Write a 5
paragraph essay about the topic, and include citations from at least 2 other resources within the
essay.
Thought Provoker: “…in a 3+1 dimensional universe, and never thought it was the least bit
curious. Their relativistic spacetime continuum was all they could see, all they could hear, all
they could feel. To them it was all there was. As long as nothing disturbed that perception, it
was true.” (Flatterland, Stewart) React to this statement.
Project: Work on your group magazine. Note that group members may want to include some of
their best work from this course as well as any new articles, stories, drawings, etc. that they
develop for the magazine.
Choose One Task from the list below:
Task 1: (Creative) Write a poem about some of the mathematical topics presented in Flatterland.
Task 2: (Self-reflection) Many of the topics presented in this novel are really interesting, but
tough mathematical topics. Choose two topics and generate a list of 5 questions for each topic
that you would like to learn as you develop a stronger math background.
Task 3: (Creative) Take an area of math you are comfortable with (like fractions, decimals,
square roots, graphing lines, etc.—things you have seen in your own math courses) and write 3
paragraphs explaining the topic, in a fun way similar to the humor and attitude Stewart writes
Flatterland with. Your audience here is a third grader, so make it entertaining and accessible for
that age group.
Week 9
Weekly Theme: Sharing the Knowledge
Reading: Finish Flatterland by Stewart.
Journal: Write a reflection about what you gained by being enrolled in this course. What did
you most enjoy and least enjoy? What can be modified for future students enrolled in this
course?
Project: Work on your group magazine. All your personal items should be printed and prepared
for the face-to-face meeting. You will only have time to put all the pages in order with your
group and figure out what you want to share with the class about your experiences putting the
magazine together basically over the Internet and what you learned about math. All the
magazines will be shared with the entire class. Make sure that you print enough copies so that
you can create a whole magazine for your entire group (and a copy for me).