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In four-dimensionalgeometry, a 16-cell is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C_{16}, hexadecachoron,^{[1]} or hexdecahedroid.^{[2]}
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes, and is analogous to the octahedron in three dimensions. It is Coxeter's $\beta _{4}$ polytope.^{[3]}Conway's name for a cross-polytope is orthoplex, for orthant complex. The dual polytope is the tesseract (4-cube), which it can be combined with to form a compound figure. The 16-cell has 16 cells as the tesseract has 16 vertices.
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Math 2B. Calculus. Lecture 01.
How Quantum Biology Might Explain Life’s Biggest Questions | Jim Al-Khalili | TED Talks
Good afternoon and welcome to the fall quarter
This is Math 2B: Calculus
and my name is Natalia Komarova
I am a professor here at the math department
and I’ll be teaching you this subject
first I want to go over the basics
of this class and tell you a little bit about
how it’s organized
first of all, I have created a website
with all the information already contained
on the website pages
but today I will still spell everything out
for you
so first of all, the textbook
so this is probably the most confusing part
of the whole thing
so it’s called calculus: Early Transcendentals
so it’s 7th edition
of calculus
by Stewart
so the most important part
is Early Transcendentals
there is a 7th edition that does not contain
these words
look at your book,
if it doesn’t say that, that’s the wrong
book
now there’s various shapes and forms of
this book
for instance, what I have only contains single
variable calculus
so that’s good for 2A and 2B
and so that will be fine for this class
there’s also a bigger book
that has, you know, parts that pertain to
other classes
like if you’re planning to take 2D or 2E
you should get the full book. The big one.
There’s also another online book,
So there are many questions that people ask
me over email
Is it okay to use a second hand copy?
And the answer is yes. It’s okay.
If you bought your book from a friend, if
it’s used,
And your electronic stuff doesn’t work
If you cannot create an account, that’s
okay
All you need from this book is the chapter
material
And the homework problems
Okay? And so I do not require you to buy a
new copy, of course
a secondhand copy is okay
any questions about the book?
Okay.
So now exams.
We’re going to have two midterms
On October 18th, and November 8th
And we are going to have a common final exam.
Question?
Will we need to bring our book to class?
No, absolutely not.
So common final
which is held on Saturday, December 7th
from 1 to 3
so I will tell you a little bit about the
common final
but first I know that according to the policy
of the mathematics department
you are not allowed to use books, notes, or
calculators during any tests.
So it is a closed book test, no cell phones,
no calculators are allowed.
Now what’s a common final?
If you’ve taken 2A, you know what that is.
It means that you come here on Saturday.
And it’s a common final that’s held across
all the sections of this class
So there are unique requirements for everybody
You have to produce your valid UCI ID card
For the midterms and the final
You have to make sure that you have an ID
And you also have to make sure that you are
recognizable in the picture
so very often students produce something that
looks like this
with just the circle instead of a face
so make sure that you get your ID replaced
such that the picture is recognizable
so another thing about common final is that
if you cannot make it
you should let us know early on
and you don’t let me know,
you let the secretaries of the math department
know
you have to fill out the form that is contained
online
you have to follow the link on the website
there is a special form
a standard form that you fill out
and there should be no problems there.
You can arrange for a make-up
But of course you have to have a valid reason
Not to attend the final
Questions about the final?
Question?
Do we need a scantron or a bluebook?
No, nothing like this. I will provide a paper
copy of the exam
And all you need is a pencil and an eraser
Yes?
When you said no calculator, are we allowed
to bring a simple calculator
no.
no but we always make sure that you can do
all the math in your head
there will be nothing horrible
more questions?
Will a calculator be needed for the class?
No, no.
You can use it when you do homework, of course
But in class, no
Okay. So now
some other assignments that you have
apart from the midterms and the final
we will have homework, okay?
so the way it works in this class is homework
is optional
which means that it’s not graded
nonetheless, this is probably the most essential
part of this class
because everything that you’re tested on
is based
on this optional homework.
So the list of homework problems is provided
in the website for the class.
And it goes by section number
So for instance section 6.1
and it gives you a list of homework problems
so you sit down and do as many as you can
after we have covered the material
if you’re fine with all the homework problems
for each section
you’ll get an A+
okay? If you’re fine with most of them
you’ll get an A. And so on.
So this is your way to study.
Do the homework problems
nobody will test them directly,
but we will have quizzes
quizzes are held once a week
at the discussion sessions
on Thursdays
and the quizzes are completely based on the
homework assignments
for the previous week
so if you’ve done your homework
you will know how to do the quiz problems
it’s either just a homework problem taken
from the list or something that is very very
close to it
so in order to prepare for the quiz, you have
to do the homework.
And the quizzes are graded
okay, so this is graded
now another part of assignments is webwork
can you raise your hand if you know what the
webwork is?
Do you know what it is?
Oh, so I see some of you are not familiar
with what it is
so this is an online homework assignment
so on the website for the class
I created a link that takes you to the webwork
homepage
this page is not active as of now
they will activate it in about two weeks
when the first assignment is posted
so the first assignment will be posted the
tenth of October
okay so until then you don’t have to worry
about it
so you go there, you log in,
and you do your problems.
They will be 8 webwork assignments during
the fall quarter
There will be every week, posted on Thursday
and due the next Friday
The first webwork assignment is based on the
problems from the beginning of the class
And they are cumulative such that
later on webwork problems could test your
knowledge from, you know, far long ago.
So everything that you have studied up to
that date
can be tested with webwork
each assignment has a varying number of problems
and I posted the full schedule of all the
webwork assignments
that is already known for class
so you will know the due dates and the dates
when these things are posted
you have an extension for thanksgiving
for thanksgiving week, you have a little bit
more time to finish
so there are a few things that you need to
know
about webwork
so among the class files
I posted a PDF file
that tells you
about webwork.
So one new thing that they told us about this
year
is that somehow it doesn’t work very well
from
so campus. It works very well from campus
if you’re away from campus, you have to
use VPN
to connect, and the instructions that will
help you set it up
are given on the website for the class.
And there are various quirks associated with
webwork
Because sometimes you may type something in
the wrong font
And it thinks that you made a mistake
all these things
please read what I posted
it gives you a lot of information
and so if you experience technical problems
with webworks,
please do not email me
because I will not be able to help you
I can only confuse you if you ask me technical
questions
About the website setup and stuff
email them
there is an email address provided on the
website of webwork
and they will be able to resolve your technical
issues
if you have a problem with your mathematics
then I’m your guide. Okay?
Email me or come to the office hours
and I will be able to help you with any math
problems
but not with technical problems and website
problems like that
questions?
Okay. Grade
consists of 40% final
20% each of each of the two midterms
10% webwork
And 10% quizzes
So the lowest quiz and the lowest webwork
are dropped
So one lowest dropped
And here the same
because of this policy, there is no opportunity
to make-up
for quizzes or webwork
there’s one seat here if you need to sit
down
and there is also one over here
there are also two seats over there
in the middle
so if you miss a quiz
or if you miss a webwork
don’t worry about it, because
because of this
you have a chance to miss one
question
so is the webwork like an online quiz, or
is it like an online homework assignment?
it is like an online homework assignment
I think you have an infinite number of attempts
More questions?
Will we be able to use a calculator on the
webwork?
oh, yeah. You’ll be doing it at home
but not during quizzes
so quizzes are like exams
more questions?
Okay. Now I think the last thing is this
so Early Transcendentals
so what does it mean?
It means that if you took calculus 2A
Prior to fall 2012
You will have some catching up to do
So the mathematics department has changed
the syllabus
We used to teach things like logarithm
Or arcsin
or e^(x)
we used to teach that in 2B
now these things are taught in 2A and you
are supposed to know them
you are supposed to know these things
and you are supposed to know their derivatives
you should know what they look like,
you should be able to plot them
so how many of you have not seen those before
okay, very good. So that’s good
however if you want to refresh your memory
by
these functions
there is some online video material
you can watch the videos
and refresh your memory about these things
such that you are well prepared for what we
have here
instead of these things, we will be studying
sequences and series
which are much more fun.
I’ve taught both, this is better
Questions?
Okay
question
are the grades going to be curved?
no, so there is no curve
there’s going to be a standard grade
so the finals are graded across the whole
you know, all the sections
then if they think that
it was too hard or too easy
then they are going to uniformly
add some points or subtract some points
like multiply by a factor
so everybody gets the same treatment
and then after that there’s no curving.
Also there is no extra credit
There’s nothing you can do
But do the homework
Okay? So everything is very straightforward
in this class
Old information is in the homework
If you have questions, if you don’t understand
something
Come to my office hours
which I will announce next week
I don’t know yet
when my office hours are
and that will be fine. Question?
You said that if the average is low, is the
final curved or the class is curved
Like overall
so if they think that the common final was
too hard,
they are going to kind of boost that grade
just for the final though?
just for the final.
So you don’t curve this class?
No. And they are not supposed to.
What’s the website?
I think you can find it on EEE
there’s a link
also you can go to my homepage and there’s
a link there
but it’s easier to go from the EEE
just to make sure, you said the webwork is
due Friday? That’s the following week, correct?
yes, it’s posted on a Thursday
it’s due the Friday next week
so it’s the following week, yes
so you have a week and one day to finish it
more questions?
Okay.
and you can always ask me questions during
the class
just raise your hand, okay?
so now we do some review
of some essential stuff that you learned in
2A
if you cannot see something, please let me
know
so I avoid some parts of the whiteboard
So today we cover section 4.9
antiderivative
okay ready?
So let’s suppose that the derivative
of function F
is given by lower case f
for all values of x in an integral
then F capital is called the antiderivative
of f.
so for example
let’s suppose that f(x) is equal to x^(2)
what’s the antiderivative of this function?
So there are two opposite operations
you can take the derivative
and you can go back and take the anti-derivative
question?
So this happens to be one of the antiderivatives
Of this function
In fact, how many are there?
Infinitely many
Plus C
So by adding a constant C
We create an infinite family of functions
each of which is an antiderivative of this
how can we check?
So this is antiderivative
To check, we say what is
F’ (x)
we have to differentiate
so we have (3x^(2) /3)+0 is x^(2)
check
because it coincides with my original function
question
I cannot see that
you cannot see this?
That’s very bad. Okay.
so are you telling me that you can only see
above this line?
okay.
so we have a theorem
if F is an antiderivative
of f
then F+C is the most general antiderivative
and here C is a constant
okay
so let’s practice and do some examples
so let’s suppose that
f(x) is cosx.
Find F(x)
so basically we are looking for a function
whose derivative is equal to cosine
so what is F(x)?
very good
sinx+C
another example
f(x) is given by x^(n)
where n is not equal to minus 1
so may I ask you please do not talk during
this class
if you have questions ask me, but do not talk.
So we have a rule to evaluate the antiderivative
of this
It’s called the power rule
power rule
okay so it tells us
it tells us that the antiderivative
is given by
this function
plus C
and in fact this is the rule that we used
here
to calculate the derivative in our very first
example
so I have a question
why is it that we have to require n
is not equal to -1
what happens to this formula when n equals
-1?
Yes. You divide by zero
So this formula is obviously not applicable
Something else goes on there
so that’s my next example
a very special case
where n is equal to -1
f is 1/x
what’s the antiderivative?
Natural log of x
In fact, it’s like this
Natural log of the absolute value of x
plus C
and we actually have to say that this formula
holds for any interval that does not contain
0
so if I give you
if I give you an interval from one to five
first
on this interval
any such function with a constant C
will serve as an antiderivative of the log
if I want to write down the most general antiderivative
on the whole
real line,
it’s something slightly more complicated
so f(x) is ln(x) +C
for positive values of x
and this logarithm of –x
oh—I’m sorry
this is noted C¬1
+ C2
When x is negative
and the antiderivative is not defined for
x=0
because the original function is not defined
for x=0
question?
can C1 and C2 be equal?
they can be equal,
but in general they don’t have to be
so this is the most general form of the antiderivative
of the function 1/x
so this function can have a discontinuity
it goes like lnx, ln(-x),
but here, when you equal x=0,
you can have a different constant.
So it’s in general a discontinuous function.
We also have something like this
For negative values of n
So let’s suppose
That f is x^(-4)
So now we have a very similar situation
So by using the power rule
My power is -4
So I have to
So x^(-4+1)
-4+1
It’s x^(-3)
Divided by –x
So the most general
antiderivative is given by the following
discontinuous function
so the antiderivative again is not defined
for x=0
because the function itself is not defined
there
and the experience
it’s discontinuating, as it goes to 0
because we’re allowed to take different
constants for negative and positive values
of x
so what we need to know for this class
is a list of antiderivatives
it’s best to know those by heart
so we have a table
in the textbook
that goes like this. We have a function
and we have a particular antiderivative
so what are the most common
functions that you should know by heart?
So first I list
a rule
if you have a function f, any function f multiplied
by a constant C
so the antiderivative also gets multiplied
by C
you know that, right?
And similarly
with the summation
the antiderivative of a sum
is the sum of two antiderivatives
now
particular examples of functions
we’ll list some of them
here I do this for completeness
so I’ll continue this table here
let’s do cosine x
gives me sinx
so this is f, this is F
so this is sinx
-cosx
secant squared
gives me tangent
secant tangent
gives me secant
these two follow from the definition of the
derivatives of secant and tangent
we know if the derivative of this is equal
to this
then the antiderivative of this
is this
this table works both ways
to go from here to here you have to take a
derivative
to go from here to here, you have to take
the antiderivative
what else?
here
we have
something that is associated with arcsin
and arctangent
so this is the best one
why is it the best one?
It’s equal to itself, right?
it's the easiest one to remember, the exponent
is equal to its own derivative
also its equal to its own antiderivative
questions?
So let’s practice
So a simple common problem
That we encounter is
find the antiderivatives of functions
so find all functions g
such that
g prime is
5+cosx
+ 3x^(2)
So look at this function
So we go to the table and of course we don’t
find that function in the table
However, if we simplify this function we’ll
find its components
in the table, right?
so the first thing we do here is simplify
and then we’ll be able to use the table
so I’m going to say that this is 5
plus cosx
and here I divide through by x, so I have
3x + x
-1/2
Now each of the components here
And we found them in the table
and I’m going to use the table
so G
so I’m sorry G
the derivative of G is this
so therefore I have to find the derivative
the antiderivative
so it’s 5x
+ sinx
how did I get the first term?
we will learn how to integrate in this class
but we have to find an antiderivative
of a constant. So
where can you find that?
for example,
here if n equals 0
that’s a constant 1, right?
So n equals 0 gives me x to the power of 1
Divided by 1, so that’s x
so the antiderivative of 1 is x
and here this tells me that multiplication
by a constant jus carries through
so this number 5
appears in front of the antiderivative
sine is the antiderivative of cosine
just pulls right from the table
this one is easy
again, it’s a power function, so
it’s 3x^(2) over 2
this one is also easy
because you use the same rule. Power rule.
Now we don’t have an integer power,
the power is equal to -1/2
so we have x^(1/2)
divided by ½
and then
don’t forget +C
this usually costs one point
on any test
questions?
do we have to simplify the ½ to a radical
or can we leave it
when it’s easy, like this
probably
I wouldn’t take a point off for this
but different graders are different
okay the next question
is a little bit more sophisticated
we can talk about differential equations
in itself it’s a huge topic
and there are whole courses taught on this
but I will just show you what it is
so the problem is like this: find f
if f’ is equal to x^6 and f(1) is equal
to 3
so I need to find the function f given this
information
two pieces of information
the first piece of information pertains to
its derivative
and the second one tells me what the value
of the function f is
at one point
x is equal to 1
so that’s what I need to find. So from this
equation
I can find f
By looking at the most general antiderivative
So general antiderivative
I take the antiderivative of x^6
Which is x^7 over 7 +C
So I found a whole bunch of functions f
they all differ by this constant
and that is why I’m given this second condition
this condition will help eliminate most of
these
and 0 on the relative one
so use f(1) equals 3
how do I use them?
I plug it in. Exactly.
So I go f(1) is 1^7 over 7 +C
and that’s supposed to be equal to 3
so I can say that 1/7 +C equals 3
where C equals 3-(1/7)
which is 20/7
therefore my function f
not the most general one, but
the actual one that solves both of these,
okay,
that’s given by x^7 over 7
plus 20/7 or ((x^7)+20)/7
so out of all of these functions, I identified
the one that satisfies not only the first
equation, but the second one, too
questions?
okay
so now we will refresh our memory with regards
to graphic antiderivatives
and we will talk about the notion of velocity
so
let’s suppose that the function f
is given graphically
something like this. One second.
So it starts off here,
goes negative,
like this,
and like this
okay, something like this
let us sketch the graph of the antiderivative
so no formula there given
and I want to draw F capital
so how do I do this in principle?
This is the derivative of this function
Now remember, what is an antider—
what is a derivative?
The derivative is the rate of change
It’s the rate of change
It's the rate at which the function changes.
if we think of the independent variable
as time, the derivative is how quickly that
function changes.
it tells you the slope, or the rate of change
and the rate of change can be interpreted
as a velocity
so let’s suppose that this is velocity of
motion
and as you can see, as time goes by, it changes
sometimes it will go faster, then it will
slow down, okay,
at this point, the velocity is equal to zero
and here it becomes negative. Which means
that
we go backward
then again at this point, we turn and start
going forward
so by using this information, I am going to
draw the position, given the velocity
okay? I have to recreate the position
given the velocity
I start with some arbitrary point
okay, let’s suppose that we know we start
at 1
and now, so look
the velocity here is positive
which means that I am going forward
I go forward means that my position, the coordinate
of my position, increases
So for a while
between time equals 0 and time equals 1 I
go forward
slower and slower and slower, but I move forward
this is my positive direction, according to
increases
at this point I stop
at this point my derivative is equal to zero
which means that I’m going to have a maximum
here, right?
and now my velocity becomes negative
at this point I start going backward
and that’s exactly what I’m drawing here
I start going backward
Faster and faster and faster
at point 2, my velocity is the fastest
negative
and then it becomes slower and slower and
slower.
so at point 2 I get something like this
and I stop at point 3
because my velocity again is 0
after 3, I continue to go forward
so my coordinate increases
and then the velocity decreases
so somehow I level off
I start going slower and slower and slower
And eventually almost stop but I don’t quite
stop
Questions?
so you should be able to take the graph of
a function
and draw its antiderivative
but I want you to think about velocity
I want you to think if this is positive,
This increases
If this is negative,
this decreases
if this is 0, it means I experience either
a maximum or a minimum
I don’t change at that point
Questions?
very good so now in the last problem
I think it’s the hardest of all
we will talk not only about velocity but also
acceleration
because they’re both connected to derivatives
and antiderivatives
so the problem is like this
suppose that
the acceleration of a particle
is given by this function
so here is my vocabulary
a is acceleration
v is velocity
and s is position
these are the common notations
and you know
that the velocity is the derivative of the
position
and the acceleration is the derivative of
the velocity
do you know this?
Okay. So what is given is the acceleration
and also some information about the position
at the beginning, the position is 2
and the velocity at the beginning is -1
find the position as a function of time
given this information
I start by saying that acceleration
is a derivative of the velocity
so v’
okay
is t+1
if I know the derivative of the velocity
I can find the velocity
by taking the antiderivative
so v(t) is found by calculating the antiderivative
of this function
which is really easy
t^2
+t +C
so given the acceleration
so let me just
I don’t want to misstep
so this is a
a is the same as v’
and it’s given by t+1
so if I know v’ I know v
and it’s given by this
unfortunately I have this unknown constant
here
but I can fix that by saying that
oh, look
v(0) is equal to -1
that tells me find the appropriate C
so we use v(0)
is -1
so what is v(0)?
it is 0/2 + 0 + C
and it’s supposed to be equal to -1
therefore C is -1
therefore v is (t^2)/2
+ t -1
so I completed the first step
using the information about the acceleration
I found the velocity.
and I also used this condition about the initial
velocity
that helped me identify this particular constant,
C
okay, so that’s step number one
step number 2, I know the velocity now
but I need to know the position
so step number 2
I go here
The velocity is the derivative of the position
So v is the same as s’
and that’s given by this formula that I
just derived
t^2 over 2
+ t - 1
From this, I can find the position
by taking the antiderivative
so if I know the derivative of s,
I can find
the antiderivative
Again I have an unknown constant called A
So what’s this constant? That’s the last
step
I’m going to use this piece of information,
the initial position
S(0) is given by 0/6 + 0/2 – 0 + A is supposed
to be equal to 2
Which means that A equals 2
So I can write down the answer
s(t) is given by t^3 over 6
plus t^2 over 2
minus t plus 2
so this is the formula that defines the position
of that particle as a function of time
questions?
Okay, thank you very much
The eight vertices of the 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs.
The Schläfli symbol of the 16-cell is {3,3,4}. Its vertex figure is a regular octahedron. There are 8 tetrahedra, 12 triangles, and 6 edges meeting at every vertex. Its edge figure is a square. There are 4 tetrahedra and 4 triangles meeting at every edge.
The 16-cell can be decomposed into two similar disjoint circular chains of eight tetrahedrons each, four edges long. Each chain, when stretched out straight, forms a Boerdijk–Coxeter helix. This decomposition can be seen in a 4-4 duoantiprism construction of the 16-cell: or , Schläfli symbol {2}⨂{2} or s{2}s{2}, symmetry 4,2^{+},4, order 64.
The 16-cell can be dissected into two octahedral pyramids, which share a new octahedron base through the 16-cell center.
As a configuration
This configuration matrix represents the 16-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 16-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
The 16-cell has two Wythoff constructions, a regular form and alternated form, shown here as nets, the second being represented by alternately two colors of tetrahedral cells.
One can tessellate 4-dimensional Euclidean space by regular 16-cells. This is called the 16-cell honeycomb and has Schläfli symbol {3,3,4,3}. Hence, the 16-cell has a dihedral angle of 120°.^{[4]} Each 16-cell has 16 neighbors with which it shares a tetrahedron, 24 neighbors with which it shares only an edge, and 72 neighbors with which it shares only a single point. Twenty-four 16-cells meet at any given vertex in this tessellation.
A 16-cell can be constructed from two Boerdijk–Coxeter helixes of eight chained tetrahedra, each folded into a 4-dimensional ring. The 16 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex. The purple edges represent the Petrie polygon of the 16-cell.
Projections
Projection envelopes of the 16-cell. (Each cell is drawn with different color faces, inverted cells are undrawn)
The cell-first parallel projection of the 16-cell into 3-space has a cubical envelope. The closest and farthest cells are projected to inscribed tetrahedra within the cube, corresponding with the two possible ways to inscribe a regular tetrahedron in a cube. Surrounding each of these tetrahedra are 4 other (non-regular) tetrahedral volumes that are the images of the 4 surrounding tetrahedral cells, filling up the space between the inscribed tetrahedron and the cube. The remaining 6 cells are projected onto the square faces of the cube. In this projection of the 16-cell, all its edges lie on the faces of the cubical envelope.
The cell-first perspective projection of the 16-cell into 3-space has a triakis tetrahedral envelope. The layout of the cells within this envelope are analogous to that of the cell-first parallel projection.
The vertex-first parallel projection of the 16-cell into 3-space has an octahedralenvelope. This octahedron can be divided into 8 tetrahedral volumes, by cutting along the coordinate planes. Each of these volumes is the image of a pair of cells in the 16-cell. The closest vertex of the 16-cell to the viewer projects onto the center of the octahedron.
Finally the edge-first parallel projection has a shortened octahedral envelope, and the face-first parallel projection has a hexagonal bipyramidal envelope.
4 sphere Venn Diagram
The usual projection of the 16-cell
and 4 intersecting spheres (a Venn diagram of 4 sets) form topologically the same object in 3D-space:
Symmetry constructions
There is a lower symmetry form of the 16-cell, called a demitesseract or 4-demicube, a member of the demihypercube family, and represented by h{4,3,3}, and Coxeter diagrams or . It can be drawn bicolored with alternating tetrahedral cells.
It can also be seen in lower symmetry form as a tetrahedral antiprism, constructed by 2 parallel tetrahedra in dual configurations, connected by 8 (possibly elongated) tetrahedra. It is represented by s{2,4,3}, and Coxeter diagram: .
It can also be seen as a snub 4-orthotope, represented by s{2^{1,1,1}}, and Coxeter diagram: or .
The Möbius–Kantor polygon is a regular complex polygon_{3}{3}_{3}, , in $\mathbb {C} ^{2}$ shares the same vertices as the 16-cell. It has 8 vertices, and 8 3-edges.^{[5]}^{[6]}
The regular complex polygon, _{2}{4}_{4}, , in $\mathbb {C} ^{2}$ has a real representation as a 16-cell in 4-dimensional space with 8 vertices, 16 2-edges, only half of the edges of the 16-cell. Its symmetry is _{4}[4]_{2}, order 32.^{[7]}
In B_{4}Coxeter plane, _{2}{4}_{4} has 8 vertices and 16 2-edges, shown here with 4 sets of colors.
The 8 vertices are grouped in 2 sets (shown red and blue), each only connected with edges to vertices in the other set, making this polygon a complete bipartite graph, K_{4,4}.^{[8]}
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN978-0-471-01003-6[1]
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_{n1})