*Zhaorui Xu*

## Full Map of Mathematopia

Adventurer’s Guide Part 1: Logic, Analysis and Calculus

Adventurer’s Guide Part 2: Linear Algebra and Groups

Welcome back to our adventure through the landscape of maths. This time we will visit the remaining parts of the map: the Land of Geometry and Probability Place.

**Geometry**

The Land of Geometry is a sacred territory that was worshiped by many great minds in Ancient Greece. Euclid the Prophet revolutionised the field by setting up an axiomatic system, based upon which all the propositions in geometry were supposed to have a rigorous and watertight proof. Influential reforms have taken place throughout history, including the outstanding development of René Descartes’ pioneering the use of coordinates to represent the location of points with the Cartesian plane.

In the two thousand years that followed the invention of the Euclidean axioms, the impeccability of the system was almost without question. However, in the 19th century, innovative voyagers discovered the Continent of Non-Euclidean Geometry where an axiom different to Euclid’s parallel postulate is taken as creed, but the same level of rigour is maintained. Sceneries on this continent are very exotic and it is not uncommon for them to oppose our intuition: for example in elliptic geometry where two parallel lines intersect!

As newcomers, your adventure in the first year will not go that far. You will carry *Cartesian coordinates* as your working equipment to survey landscapes like the beautifully curved Brooks of Conics.

At the edge of the river bank where the Land of Linear Algebra can be seen in the distance, you will gain an insight into the close relationship between geometry and linear algebra. You will learn that *rotations* and *reflections* of shapes can be concisely represented by matrices, and more precisely, *orthogonal matrices* which preserve lengths or distances.

**Probability**

Probability Place is a very cosy land to the South of Mathematopia filled with sparkles of playfulness. For most of your first year here, you will be exploring different types of distributions such as the Poisson distribution, the binomial distribution, and the Bernoulli distribution, which describe *discrete random variables*; as well as the normal distribution, and the exponential distribution which describe *continuous random variables*. Important concepts that you will uncover are their *expectations* and *variances*.

You will also be introduced to some more sophisticated and useful tools in the form of *probability generating functions*. Each one a single function that encapsulates all of the information about a specific distribution – for this reason they are often referred to as the genes of a distribution.

On the grassland you will see sheep strolling leisurely, taking their steps seemingly at random. With the knowledge of *linear difference equations*, you will be able to calculate the probability of a sheep either reaching the river on one side or ending up by the hills on the other.

Even trees here have distinct regional characteristics with branches extending almost indefinitely in all directions. The *branching process*, as the trees’ form implies, can be used as a model to predict progress of real life events. For example, it can predict whether a population will expand or die-out through generations and helps to quantitatively reveal the likelihood of these occurrences.

Finally, when coming to random samples, you will learn the *Weak Law of Large Numbers* (in the years to come, the* Strong Law of Large Numbers *will be unveiled), as well as some rules of living summarised by your adventuring predecessors. For instance,* Markov’s Inequality* and *Chebyshev’s Inequality*.

**Non-maths Area**

For the most adventurous of warriors who wish to stray beyond the borders of Mathematopia, you may wish to learn of an activity known as ‘punting’ – popular in both Oxford and Cambridge. A punt is a small, narrow boat designed to be used in shallow rivers, and the punter uses a long pole to navigate and to power the boat by pushing on the river bed. However, there are small differences in the tradition at the two locations: in Oxford the punter usually stands inside the boat with the till (‘deck’) of the boat facing forwards, while in Cambridge the punter stands on the till at the rear of the boat. The rivalry between the opposing tribes is fierce, so be sure to tread carefully if crossing the boundary…

The full ‘Map of Mathematopia’ can be found here.

Part 1 of the Adventurer’s Guide to Mathematopia can be found here.

Part 2 of the Adventurer’s Guide to Mathematopia can be found here.

[…] Part 3: Geometry and Probability […]

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