Brothers, cousins, and foes - what am I?

Question

I am one of many siblings in a great family.
We are the most perfect of our race,
Because each of us has a clear worst enemy.
My foe - I know him well - is much like me,
But we continually fight each other;
We never hide our hatred for each other.

Wave at my eldest brother and he becomes me.
When my dear cousin finds the end of her life,
She too can turn into me, in her gentle way.
My younger brother thinks too highly of himself,
So smooth and perfect that he'll always outstrip me.
Often he even steals my words, confusing all concerned.

What am I?

Clarification:

1) As @KeithS worked out, the answer is something in pure maths, and more abstract than anything anyone has tried so far. You need to know a bit of undergraduate-level maths, at the level of basic topology and group theory.

2) Every line has a thought-out meaning, so you need to find a solution that matches everything from "we never hide our hatred for each other" to "he'll always outstrip me". I could give you a word for each of the "family", "siblings", "cousin", "eldest brother", and "younger brother". About the only red herring is "in her gentle way", which is just a nod to Thomas Hardy.

Hint:

The answer is a mathematical object which someone has mentioned (not proposing it as an answer, but just mentioning it in passing) in one of the responses below. I won't say which one!

New hint:

I am the only one who can go directly to Leipzig from a well-policed city.

Answer 1

I am

a homeomorphism. i.e. an isomorphism between two topological spaces. I am continuous and so is my inverse.

My family is

the set of invertible functions. Our clear worst enemy is the inverse function.

And my race is

the set of functions.

We are perfect among our race because

we are invertible.

My eldest brother is

a noncontinuous invertible function. "Wave" at my eldest brother means to make him continuous, and thus become me. (sine waves are continuous)

My cousin is

a homomorphism. When she "finds the end of her life" (the letter e), and adds it to herself, it forms my name.

My younger brother is

A diffeomorphism. i.e. an isomorphism over a differentiable manifold. It is a smooth function whose inverse is also smooth. Because smoothness is a higher restriction than continuousness, the younger brother outstrips me. However, people are often confused about the difference between smooth and continuous.

Answer 2

Partial answer.

It's a proton.

We are the most perfect of our race, Because each of us has a clear worst enemy.

Every particle has an anti-particle

My foe - I know him well - is much like me, But we continually fight each other; We never hide our hatred for each other.

This would be the anti-proton

Wave at my eldest brother and he becomes me.

If you hit a neutron with the right wavelength, you can strip off a sub-particle and convert it into a proton

When my dear cousin finds the end of her life, She too can turn into me, in her gentle way.

Something to do with nuclear decay

My younger brother thinks too highly of himself, So smooth and perfect that he'll always outstrip me. Often he even steals my words, confusing all concerned.

an Electron? in a high energy orbital?

It almost fits.

Answer 3

I think it is

Trig functions

I am one of many siblings in a great family. We are the most perfect of our race, Because each of us has a clear worst enemy.

Worst enemies: Sine -> Cosecant, Cosine -> Secant and Tangent -> Cotangent. Perfect: describe a circle.

My foe - I know him well - is much like me, But we continually fight each other; We never hide our hatred for each other.

Sine vs Cosine.

Wave at my eldest brother and he becomes me. When my dear cousin finds the end of her life, She too can turn into me, in her gentle way.

Sine turns to cosine or Secant and Cosecant.

My younger brother thinks too highly of himself, So smooth and perfect that he'll always outstrip me. Often he even steals my words, confusing all concerned.

Tangent, not bounded on the y axis. Tan = Sin/Cos (Steals my words)

EDIT

Second try at this.

I think it is:

f(x) = x^z; z is all reals. exponential functions

I am one of many siblings in a great family. We are the most perfect of our race, Because each of us has a clear worst enemy.

Great family: a family of functions. Worst enemy: if y = x^z then enemy is x = y^(1/z). The inverse

My foe - I know him well - is much like me, But we continually fight each other; We never hide our hatred for each other.

foe/enemy is much like me: inverse functions have very similar properties to that of the original function. But we continually fight each other: inverse functions are the same graph but with the axis changed.

Wave at my eldest brother and he becomes me. When my dear cousin finds the end of her life, She too can turn into me, in her gentle way. My younger brother thinks too highly of himself, So smooth and perfect that he'll always outstrip me.

I think this is talking about the integral and derivative.

Often he even steals my words, confusing all concerned.

Same family. could both be called exponential functions?

Answer 4

What about...

An integer

I am one of many siblings in a great family. We are the most perfect of our race, Because each of us has a clear worst enemy.

An integer is a whole number ('perfect') with a clear worst enemy, its negative value.

My foe - I know him well - is much like me, But we continually fight each other; We never hide our hatred for each other.

For every integer, the opposite integer exists, 'fighting' each other.

Wave at my eldest brother and he becomes me. When my dear cousin finds the end of her life, She too can turn into me, in her gentle way.

A 'waving' motion in real life looks similar to a '-' or minus sign, so it could just simply be saying 'subtract from my older brother and he becomes me', which would be true, assuming the 'older brother' is just a higher integer. For the cousin one, the character mentions he is one of many siblings (a brother or sister) in a great family, so the cousin may be referring to a non-integer floating point number. She 'turns into' an integer by rounding up or down.

My younger brother thinks too highly of himself, So smooth and perfect that he'll always outstrip me. Often he even steals my words, confusing all concerned.

The younger brother is the integer 0 ('smooth and perfect'), and thinks highly of himself because he is his own opposite. He steals the 'words' of other integers by taking their beginning numbers (1, 3) and adding to them (10, 30), 'stealing' part of their values.

Answer 5

I am one of many siblings in a great family.

The narrator is x -> e ^ x (e is the Euler–Mascheroni constant), and the family is the set of all exponential functions, x -> a ^ x, a is real.

We are the most perfect of our race,

Race means the set of all functions. Perfect because of the nice properties about derivatives also being exponential functions.

Because each of us has a clear worst enemy.

Worst enemy means inverse, in this case (natural) logarithic function.

My foe - I know him well - is much like me,

x -> log(x) has many similar properties to the exponential function.

But we continually fight each other;

Both the exponential and logarithm functions are continuous.

We never hide our hatred for each other.

???

Wave at my eldest brother and he becomes me.

I think the eldest brother is x -> a ^ x where a < e. There is a transformation from the eldest brother function to the narrator function. The word wave is a reference to the fact that the exponential function can be constructed from sine and cosine "wave" functions.

When my dear cousin finds the end of her life,

I think the cousin is x -> e ^ -x. As x increases towards positive infinity, this function decreases towards zero.

She too can turn into me, in her gentle way.

There is a transformation from one function to the other.

My younger brother thinks too highly of himself,

By the same logic as the younger brother, this must be x -> a ^ x where a > e. (Probably a = 10; see below.) This function will return values greater than the narrator function for all x > 0.

So smooth and perfect that he'll always outstrip me.

The exponential functions are infinitely differentiable (smooth). Outstrip me means the same as "thinks too highly of himself".

Often he even steals my words, confusing all concerned.

"Exponential function" can refer specifically to x -> e ^ x, or to the more general family of functions. In particular, x -> 10 ^ x is often confusingly referred to as the exponential function.

Answer 6

What about

arithmetic operators

I am one of many siblings in a great family. We are the most perfect of our race, Because each of us has a clear worst enemy.

addition / subtraction, multiplication / division

Wave at my eldest brother and he becomes me.

multiplication can be expanded as series of additions

When my dear cousin finds the end of her life, She too can turn into me, in her gentle way.

power operator can too be ultimately expanded as series of additions

My younger brother thinks too highly of himself, So smooth and perfect that he'll always outstrip me. Often he even steals my words, confusing all concerned.

well, this one I am not sure, factorial maybe?

Answer 7

Here is my guess.

Equivalence relations

I am one of many siblings in a great family.

Family is the universal set, and siblings probably means those in the same class.

We are the most perfect of our race, because each of us has a clear worst enemy.

Race refer to all relations. We are perfect because we are equivalence, and being equivalence, we have an inverse relation that are "perfect" too. That, my friend, is my worst enemy.

My foe - I know him well - is much like me, but we continually fight each other;

He is indeed a perfect creature, just like me.

We never hide our hatred for each other.

Indeed, one can easily find the inverse relation when given an equivalence, nothing to hide.

Wave at my eldest brother and he becomes me.

Yeah of course, we are equivalent, so if I am x, and my brother is y, x ~ y. Awesome.

When my dear cousin finds the end of her life, she too can turn into me, in her gentle way.

I'm not so sure, maybe some clousure under certain operator. Strictly speaking, I did not have any education in sets and relations yet, so my knowledge is limited here. Tried my best.

My younger brother thinks too highly of himself, so smooth and perfect that he'll always outstrip me. Often he even steals my words, confusing all concerned.

Well, we can represent a certain partition with just a single element. Like maybe the n ≡ k modulo 3 operator, we can represents all multiple of 3, which includes 6, 9, ... 3k by [3]. It's really stealing my identity as the holy go-to person for all questions, 42.

Pretty sure this isn't correct. I feel like there is too many holes that I only breifly dealt with. Just post this because it might be one interesting answer.

Answer 8

I know this isn't going to be the answer, but just wanted to share my thoughs.

I am

Infinity

in a great family

of numbers

We are the most perfect of our race,

we can't be beaten

Because each of us has a clear worst enemy.
My foe - I know him well - is much like me,
But we continually fight each other;
We never hide our hatred for each other.

Positive Infinity vs Negative Infinity. Which one is stronger ? Impossible to tell.

Wave at my eldest brother and he becomes me.

(Positive) Infinity + 1 (or any other positive number) is Infinity (or so XD)

When my dear cousin finds the end of her life, She too can turn into me, in her gentle way.

anything (number/function) that tend to infinity...

My younger brother thinks too highly of himself,

(positive) infinity - 1 ?

So smooth and perfect that he'll always outstrip me. Often he even steals my words, confusing all concerned.

(positive) infinity - 1 is not really infinity, but is so great that it can be assumed as infinity aswell... or so. XD.

Answer 9

I think it's I know it's not

the sun

I am one of many siblings in a great family. We are the most perfect of our race, Because each of us has a clear worst enemy.

stars

My foe - I know him well - is much like me, But we continually fight each other; We never hide our hatred for each other.

Moon

Wave at my eldest brother and he becomes me. When my dear cousin finds the end of her life, She too can turn into me, in her gentle way.

At the end of life stars become other stars (states)

My younger brother thinks too highly of himself, So smooth and perfect that he'll always outstrip me. Often he even steals my words, confusing all concerned.

Young stars often burn through their energy quickly. Young being relative.

Answer 10

Sine

I am one of many siblings in a great family. We are the most perfect of our race,

The family is Trig functions

Because each of us has a clear worst enemy. My foe - I know him well - is much like me, But we continually fight each other; We never hide our hatred for each other.

The sine function is a periodic one. It can not overcome the +1 / -1 limits.

Wave at my eldest brother and he becomes me.

The cousin is the cosine: sin($\alpha$ +$\pi/2$ ) = - cos($\alpha$ ).

When my dear cousin finds the end of her life, She too can turn into me, in her gentle way.

The dear cousing is a McLaurin /Taylor series If n$\rightarrow$ $\infty$, the series is really the same as sin.

My younger brother thinks too highly of himself, So smooth and perfect that he'll always outstrip me. Often he even steals my words, confusing all concerned.

The brother is tan, thus it goes to infinity at 90° against 1. You can transform tan as $\frac{sin}{cos}$

Answer 11

multiplication

I am one of many siblings in a great family.

mathematical operations

We are the most perfect of our race,

multiplication, division, addition, subtraction are the most quintessential mathematical operations

Because each of us has a clear worst enemy.

multiplication vs. division, addition vs. subtraction

My foe - I know him well - is much like me,

multiplication is very related to division

But we continually fight each other;

during algebraic operations, division and multiplication are constantly used to manipulate the equation

We never hide our hatred for each other.

they are open and in a sense equal to each other

Wave at my eldest brother and he becomes me.

multiplication is just repeated addition

When my dear cousin finds the end of her life,

when you subtract (the dear cousin) a negative

She too can turn into me, in her gentle way.

addition is performed and can be just as easily turned into multiplication

My younger brother thinks too highly of himself,

division is multiplications enemy

So smooth and perfect that he'll always outstrip me.

division undoes multiplication

Often he even steals my words, confusing all concerned.

when a coefficient is attached to a variable, division will take the coefficient away from the variable, a letter

i feel like i am either pretty close or very off point

Answer 12

Ah, more abstract then:

We are:

a family of functions

The most perfect of our race:

well formed functions, not piece-wise or broken

Each of us has a clear worst enemy:

The functions are invertible

Wave at my older brother and he becomes me:

I am the integral of my older brother

My dear cousin is:

a limit, limits of a function taken to infinity can be the equivalent of derivatives and integrals

My younger brother is:

my own integral, he is a higher order function, he is also a smoother function, he also includes all of my own terms and adds some more of his own.

Answer 13

The Great Family is:

matrices

Me and my siblings are:

rotation matrices

We are the most perfect of our race because:

we are all invertible

Our clear worst enemy is our:

inverse

My foe is much like me, I know him well becuase:

the inverse of a rotation matrix is another rotation matrix

We never hide our hatred for each other:

as before the inverse of a rotation matrix is another rotation matrix and inverting a matrix does not lose or hide data or information

My eldest brother is:

The identity matrix.

Wave at my eldest brother and he becomes me because:

As the identity matrix he can become any matrix that is multiplied by him, and rotation matrices involve the wave functions sine and cosine

When my dear cousin finds the end of her life she too can turn into me:

I'm uncertain of this one the cousin might be a scale or shear matrix which if the numbers are just right can become a rotation matrix, there is word play though on 'turn into me' since I am in fact a rotation matrix

My younger brother is:

a matrix with 0'd elements possibly the null matrix

He is smooth and perfect because:

all of his elements are smooth perfect 0's

He will always outstrip me because:

0's applied to a matrix remove data that can never be recovered. this is the same as him stealing my words confusing all concerned.

I'm not sure why he thinks so highly of himself though.

Answer 14

Is it the

union function?

I am one of many siblings in a great family.

Which would be set operation functions

We are the most perfect of our race, because each of us has a clear worst enemy.

Each union function has a corresponding intersection function

My foe - I know him well - is much like me, but we continually fight each other;

The intersection function and the union function each undo each other

We never hide our hatred for each other.

The intersection function function which corresponds to a union function would either be defined in terms of it, it would have the same parameters. Or you might be getting at the fact the symbols for these operations are vertically symmetrical opposites?

Wave at my eldest brother and he becomes me.

He would be the identity function - the "oldest" function in terms of a functional definition from first principles

When my dear cousin finds the end of her life, she too can turn into me, in her gentle way.

This is the summation function, which can incrementally add set elements until reaching the same set membership as the union function would create.

My younger brother thinks too highly of himself, so smooth and perfect that he'll always outstrip me. Often he even steals my words, confusing all concerned.

The younger brother would be the recursion function, who would be able to defining himself in terms of the original union function. Recursion functions are just generally quite confusing!

Answer 15

This is not full answer as well, but "stealing my words" makes me think of either

1) Sets (younger brother could be a subset)

or

2) Geometric sequence (where younger brother could be a sequence that contains a subset of another sequence) - e.g. 1, 1/2, 1/4 and 1, 1/4. 1/16

Answer 16

Not an answer, rather than some random rant:

I believe that the Leipzig reference is a reference to Leibniz (not because they sound similar, rather than Leibniz being born in Leipzig, the Uni at Leipzig, etc.). So following this train of thought, we arrive at a lot of philosophy and maths. Topology and calculus, to be precise.
I have two suggestions, first - fractals, defined with functions that can have their inverse. Fractals are self-similar, which explains why age difference is a thing in the family. As each member becomes older, they get more-complex (higher-definition of sorts). That is why the smaller brother is smoother.
The second suggestion is about manifolds. A manifold can have different spatial properties depending on "scale". For example the area around each point on a sphere resembles Euclidean space when the area is small enough. Each manifold is defined by a function and the enemies are the inverse?