Malin Christersson’s Cube Toy
I was looking through the Geogebra site and found this lovely applet Orthographic Projection by Malin Christersson.
This is a lovely tool for investigating one of my favourite facts about hexagons:
The area maximizing orthogonal projection of a cube is the regular hexagon.
It turns out that Malin has tonnes of awesome geometry stuff online!
- Hyperbolic tiler!
- Pythagora’s Tree!
- A whole Tumblr full of great animations!
Awesome math art!
Basic Combinatorics.
First we recall a little bit of terminology:
1.1. Sets and functions
A set is a collection of elements . We write a set by surrounding its list elements with curly braces. For example: ,
. We also use set constructor notation
where
is some statement about
that can be true or false. For example:
,
. We write:
,
for the set of natural numbers,
for the set of rational numbers,
for the set of integers.
We write to mean that
is in the set
. We write
. We write
. We write
for
if
. If
then we say that
and
are disjoint sets.
We write for the set of ordered pairs of elements.
Definition 1 A function
is injective (one to one) if:
implies
. A function is surjective (onto) if: for all
there is
such that
. A function is bijective if: for all
there is
such that
. The number of elements in a set
is written
.
1.2. Basic formulae
The basic facts of combinatorics are very simple.
- If
then there is no injective function from a set with
elements to a set with
elements. (This is called pigeon hole principle.)
- If there is a bijective function from
to
then
.
- If
and
are disjoint then
.
- If
and
are disjoint then
.
-
.
- The number of
element subsets of a set with
elements is:
. (Why is this an integer? Prove it.)
- The number of functions from an
-element set to a
element set is
.
- If
then the number of bijective functions from
to
(permutations of
) is
.
There are a couple formulae that are handy to remember:
- There are
subset of
.
- Suppose you have
objects of types 1, 2,
,
respectively. The number of ways of arranging all the objects is:
- Suppose you have
identical objects that you want to distribute among
containers. The number of ways to do this is:
. (Why?)
Of Committees and Meetings
I’m happy to announce that I survived my PhD supervisory committee meeting. After 362 days, my committee met once again (we last met Apr 10th 2013), and things went well.
Below I’ve included some details on the process and some post-meeting thoughts. All of it is based purely on my own experience, but might be helpful to others.
2014 Mentoring
I’ve added a page about the mentoring project that I’m working on with three Gr. 12 students this semester. I’ll be updating the page as we go. For more information see Mentoring — 2014. From the introductory remarks:
The plan for the semester is an ambitious one. We’re going to understand the structure of all the regular convex polytopes in all dimensions, and build up a intuition for dimensions greater than three. We’ll spend most of our time learning the tools we need to understand how a geometric object can be pieced together. These tools will include vectors, metric spaces, symmetry groups, and simplicial complexes. I’m taking a very broad view of what constitutes a tool and counts as information about a space. The final result of the project will be a poster presentation about the solids and some 3-dimensional “nets” of the 4-dimemsional solids (the simplex, cube, cross-polytope, and 24-cell).
I can’t resist saying things about hyperbolic geometry. Therefore, if we get to the end of the proposed project, we’ll take a stab at using out high brow high dimensional intuition to understand the Gieseking manifold. There is more than enough stuff to say about the platonic solids, so we’ll see how far we get.
The Rotationally Distinct Ways to Label a Die
I’m giving a talk at the Canadian Math Camp this year. I’ll be showing the kids of how to count the number of ways to label a six sided die up to the rotational symmetries of the cube. Here is the handout for the talk with questions about dice labellings, the 15-puzzle, and permutation groups.
For the curious the labellings are below the cut. Please note that there are typos in the table below. Alex Fink kindly pointed them out and they will be fixed eventually. For now they are an exercise in keen observation.
Questions about Discs
Over the past couple weeks I’ve been asked a lot of questions about discs in Euclidean space. In this post we’ll be putting pennies on a table, refining covers of discs, and trying to cram lots of balls into high dimensional balls. Some open questions about putting pennies on tables occur below.
A Bijection
While grading an assignment on cardinality, I ran into the answer to the following problem:
Exercise 1 Show that
is a bijective map
.
The Spherical Isoperimetric Inequality
Here is an application of the spherical isoperimetric inequality.
Fact You can’t cut up a beach ball into equal parts with a path that is too short.
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