# Who Wants to Be Normal?

# BLOG: Heidelberg Laureate Forum

In mathematics, we often use terminology that has a precise meaning, but many terms are also words used outside of mathematics too – sometimes with a totally different meaning. ‘Obtuse’ means one thing when you are talking about an angle, but a totally different thing if you are referring to a person; an expression might be something you wear on your face, or write on a page.

But one word which has received more abuse than many in mathematics is the word **normal**. Chambers’ Dictionary defines it as ‘regular, typical, ordinary’ and ‘not deviating from the standard’; but in mathematics we use it for several things, only some of which fit with this definition. Here are my top normals in maths: Hopefully you will find them all to be completely normal.

**Normal Distributions**

Probably the closest in meaning to the… well, normal use of the word normal, a **normal distribution** is a statistical distribution which represents how you might expect data values to be distributed across a range. For example, if you measure the heights of a large group of people, you might expect the bulk of people to be around average height, with maybe a few outliers who are extremely short or tall. The distribution of heights overall will follow a** bell curve** – which has a peak in the middle and drops smoothly off to either side, like a bell shape.

The centre of the bell will be found at the mean value, µ (which, for a normal distribution is equal to the median and mode averages of the data), and the width of the bell is described by the standard deviation of the data, denoted σ². If the values are clustered together, the bell will be narrower, and if they are more spread out, it will be a wider shape. It is completely symmetrical around the centre of the curve, and flattens out to horizontal at each side.

The standard normal distribution – shown above in red – is the particular curve resulting from a mean of 0, and a standard deviation of 1. Every other normal distribution will be some version of this, depending on the values of the mean and standard deviation. This kind of shape turns up in all kinds of datasets – often, real-world physical data follows this type of curve.

The normal distribution is useful to statisticians, because if you are expecting to see something behaving normally, and thus creating normally distributed data, you can use it to determine if an observation you have made was likely to have happened by chance, or if (for example) a medical intervention, or experimental change you have made, has had a significant effect.

**Normal Vectors**

A less obvious use of the word normal is in describing something that is pointing in completely the wrong direction. Imagine you have a curved surface (a real one, or something more abstract), and pick a point on the surface. If you want to describe the direction at that point which is moving directly away from the surface, that is called a **normal vector**.

If your surface is a 2D shape like a circle or a curve, you can draw a tangent at that point, and construct a line that is perpendicular to the tangent – meeting it at right angles. The normal vector will lie in this line.

If your shape is more of a 3D surface, you can instead construct a **tangent plane** – a small patch of flat surface, which touches the surface only at that point, and contains all the tangent lines at that point in every direction. The vector sticking perpendicularly up out of this plane is the normal vector.

This object is useful in mechanics for defining how forces are going to act on an object, and is used in computer graphics to understand how light will reflect off an object, to help create realistic 3D renderings.

**Normal Numbers**

The word ‘normal’ is also used to describe a particular type of number – one with an infinite decimal expansion, which additionally has the property that every digit occurs with an equal frequency – we say the digits are distributed uniformly. So, if the number is written in base 10 (decimal), each of the digits 0 to 9 will occur exactly one tenth of the time in the decimal expansion.

A number is **normal in a given base b,** if every digit from 0 to b occurs with equal frequency, and every finite string of digits of a given length occurs with equal frequency too. This means if a number is normal in base 10, it is guaranteed to contain every possible finite string of decimal digits somewhere in its infinite decimal expansion. If a number is normal in every possible number base, it is **absolutely normal **(and that is absolutely actually what it is called).

The fascinating thing about normal numbers is, it can be quite tricky to find them, but also completely trivial. If I wanted to give you an example of a normal number in base 10, I could write:

0.1234567891011121314151617181920212223242526272829…

Known as **Champernowne’s constant**, this number consists of all the numbers from 1 upwards, written in order one after the other. This will, by definition, contain all the possible strings of digits, and will satisfy the conditions of being normal – but it is also pretty boring, because I designed it that way. While it is known that plenty of normal numbers exist, nobody has yet managed to show that a given number is normal unless it has been specifically constructed to have that property.

There are numbers which are conjectured to be normal – one example is π, the circle constant. It is suspected that π is a normal number, but nobody can say for sure. Does it contain every possible string of digits in base 10? It is hard to say, but you might enjoy this website I made for a particular date a few weeks after π day a while back, along with this accompanying explanatory blog post.

**Normal Subgroups**

One final use of the word ‘normal’ is in group theory – I have written about groups here before, but essentially they are sets of objects which can be combined in pairs, and the result will be an object from the same set (we say the set is **closed **with respect to the combining operation). For example, numbers form groups, since when you add two numbers together, you get another number.

Groups can be made up of numbers, permutations (which I have also written about here before) or even symmetries. Each element in the group has an **inverse** – if the element is g, its inverse is g^{-1}. For a number, this might be its negative, or for a rotation symmetry it might be a rotation by the same amount in the opposite direction. An element combined with its inverse will cancel out and have no effect.

A subgroup is a collection of objects within a group, which also has to be closed (so, combining any two things in the subgroup, the result has to stay in the subgroup). For example, take the group of symmetries of a square. Any two symmetries can be combined – by applying one followed by the other – to get another symmetry. Within this group, which has eight elements, there is a subgroup consisting of the four rotations (by 0, 90, 180 and 270 degrees) since combining any of these will only ever give another rotation.

A normal subgroup is one which satisfies a particular condition – that it is **closed under conjugation**. **Conjugating** an element, say g, involves taking a second element, h, and combining h with g followed by the inverse of h: hgh^{-1}. (Conjugation is closely related to abelian groups, which I describe in the blog post linked above).

If we take all the elements in a particular subgroup and find their conjugates with h (where h can be any element of the whole group), we will get a new subgroup. If this is the same subgroup as the one you started with, for any h you choose, then we say it is a **normal subgroup**.

This might sound like a very fiddly technical definition, but for a more intuitive way of thinking about it, imagine h is a rotation by 90 degrees. If we rotate something by 90 degrees, perform another transformation to it, then rotate it back again, it is like we have rotated the transformation around. Or, if we are working in a group of permutations, conjugating by a permutation is like relabelling all the objects in the set, performing a shuffle, then relabelling them back again.

Normal subgroups are important, as they help us to understand the structure of the group as a whole, and how elements will behave when combined. Normal subgroups can be thought of as like the prime factors of a number – a **simple group** is one with no normal subgroups, which would be analogous to a prime number. Since group structures can be seen underlying many mathematical ideas, understanding how normal subgroups interact can be an extremely useful tool.

So if you are worried about whether or not something is normal, do not worry – it probably is!

Katie Steckles wrote (29. May 2024):

>

[…] Normal Vectors […] Imagine you have a curved surface […] and pick a point on the surface. If you want to describe the direction at that point which is moving directly away from the surface, that is called a normal vector.>

construct a tangent plane – a small patch of flat surface, which touches the surface only at that point,Surely the function

\[ z_{\mathcal S}[ ~ x, y ~ ] := \begin{cases} 0 \text{ if } ((x, y) = (0, 0) \cr (x^2 + y^2)^{3} ~ \left( \text{Sin}\left[ ~ \frac{1}{(x^4 + 2 x^2 y^2 + y^4)} ~ \right] \right)^2 \text{ otherwise.} \end{cases} \]

represents a surface \(\mathcal S\) (“in” \( \mathbb R^3\), with suitable Pythagorean metric, thus providing a flat metric space with adapted coordinates),

which does have a tangent plane \(\mathcal T\) “in” its point \((x, y) = (0, 0)\) (represented by the function \( z_{\mathcal T}^{}[ ~ x, y ~ ] := 0 \),

such that (it seems, depending on the detailed notion of

“patch”) any“patch”of the tangent plane \(\mathcal T\) containing point \((x, y) = (0, 0)\) — regardless how small — touches surface \(\mathcal S\)not onlyin point \((x, y) = (0, 0)\),but alsoin many other points, too.>

and contains all the tangent lines at that point in every direction. The vector sticking perpendicularly up out of this plane is the normal vector.Therefore I wonder whether the function

\[ z_{\mathcal F}[ ~ x, y ~ ] := \begin{cases} 0 \text{ if } ((x, y) = (0, 0) \cr (x^2 + y^2)^{3} ~ \left( \text{Sin}\left[ ~ \frac{1}{(x^6 + 2 x^2 y^2 + y^6)} ~ \right] \right)^2 \text{ otherwise.} \end{cases} \]

represents a surface as well;

and if so, with the corresponding surface being called \(\mathcal F\),

whether the same plane \( z_{\mathcal T}^{}[ ~ x, y ~ ] := 0 \) does contain all the tangent lines which surface \(\mathcal F\) has at point \((x, y) = (0, 0)\), in every direction in which surface \(\mathcal F\) has tangent lines at point \((x, y) = (0, 0)\) at all, too.

p.s.

Frank Wappler wrote (29.05.2024, 18:47 o’clock):

>

[…] Surely […]Here’s a link which can be useful for checking what I had supposed:

https://www.wolframalpha.com/input?i=Plot%5B+D%5B+r%5E%286%29+%28Sin%5B+1%2F%28r%5E%284%29%29+%5D%29%5E2%2C+r+%5D%2C+%7B+r%2C+-1%2C+1+%7D+%5D

p.p.s.

Frank Wappler wrote (29.05.2024, 18:47 o’clock):

>

[…] Therefore I wonder whether […]Here are two (contrasting) links which can be useful for motivating what made me wonder:

https://www.wolframalpha.com/input?i=Plot%5B+D%5B+%282+x%5E2%29%5E%283%29+%28Sin%5B+1%2F%282+x%5E6+%2B+2+x%5E4%29+%5D%29%5E2%2C+x+%5D%2C+%7B+x%2C+-1%2C+1+%7D+%5D

vs.

https://www.wolframalpha.com/input?i=Plot%5B+D%5B+x%5E%286%29+%28Sin%5B+1%2F%28x%5E6%29+%5D%29%5E2%2C+x+%5D%2C+%7B+x%2C+-1%2C+1+%7D+%5D

Furthermore, I insist on recognizing bells of finite diameter (with cross-sectional curves such as this).