# Brothers, cousins, and foes - what am I?

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.

• If it is math, maybe the great family is numbers. That would make warring pars a number and their negative. – Travis Don Kindred Nov 18 '14 at 17:54
• @TravisKindred - Cousin is just cousin. There is a pattern to the family relationships, but not of a wordplay sort. – Rand al'Thor Nov 18 '14 at 18:01
• I thought of that, but numbers and their negatives "complement", they don't really "fight". The behaviors of the rest of the "family" are key to the riddle, I think; the elder brother is something that can easily transform into the speaker, while a cousin is something that will become the speaker "at the end of her life", and a younger brother is something smoother and often confusing. I thought of the function f(x) = 0, with other classes of function becoming indistinguishable from this one by derivation or asymptotic trending, but it doesn't really fit the description of a "younger brother". – KeithS Nov 18 '14 at 18:02
• I am thinking shapes...like spheres – stackErr Nov 18 '14 at 18:31
• I'll add some hints soon. If nobody gets the answer I intended, I'll accept the best answer I've got. – Rand al'Thor Nov 18 '14 at 20:54

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.

• Many congratulations! I'm so glad this riddle finally got solved. The only bit that doesn't quite fit is the 'eldest brother'; this was supposed to be equality (the most basic of invertible morphisms), because the symbol = turns into the symbol for homeomorphism when you add a wavy line. Also lines 5 and 6 are supposed to mean 'continuous, not discrete/discreet', and the final line is meant to refer to words like 'manifold' which can be either continuous manifolds (up to homeo) or smooth manifolds (up to diffeo). – Rand al'Thor Dec 4 '14 at 9:54
• @rand I think my bounty helped just a little :D – warspyking Dec 4 '14 at 12:08
• @warspyking - Indeed; thanks for being an Altruist! :-) – Rand al'Thor Dec 4 '14 at 12:37
• @rand When I read that you'd end up accepting the answer you weren't looking for I immediantly stepped into action. I wasn't gonna spoil someone's chance for rep, (and your joy of someone sving it) – warspyking Dec 4 '14 at 18:57

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.

• Another nice answer! Unfortunately I can't +1 for another 5 hours since I've reached my daily limit. But the bit about the younger brother doesn't really fit, and what about "But we continually fight each other; We never hide our hatred for each other"? – Rand al'Thor Nov 18 '14 at 18:42
• Could be fighting neutrons for space in the nucleus? – Raystafarian Nov 18 '14 at 18:44
• anti-particles destroy each other. – Chris Cudmore Nov 18 '14 at 18:51
• Anti-particles are most often seen in the weak interaction; quarks within nucleons are momentarily converted to anti-quarks by the mediation of W and Z bosons. However, this interaction is very well-"hidden"; it wasn't until this year that the elementary boson which proves the theory correct was isolated. – KeithS Nov 18 '14 at 19:26
• @randal'thor Protons experience a repulsive electrostatic force from other protons, their foes who are very much like them. – Jason Patterson Nov 25 '14 at 13:19

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?

• Another nice answer! Unfortunately I can't +1 for another 5 hours since I've reached my daily limit. But the bit about 'elder brother' and 'cousin' doesn't work that well. Also, is it sin vs. cosec or sin vs. cos? – Rand al'Thor Nov 18 '14 at 18:48
• I was thinking more along the lines of the inverse functions rather than the co functions – Chris Cudmore Nov 18 '14 at 18:48
• @randal'thor could interpret it both ways. sin vs cosec or sin vs cos since. sine and cosine on a graph look like one is always trying to "beat" the other in an infinte race :P – stackErr Nov 18 '14 at 18:50
• @randal'thor so is this the answer and you are just waiting for a better explanation or this is not the answer? – stackErr Nov 18 '14 at 18:57
• @stackErr - Your second answer is similar to Richie Cotton's. Nice, but still on the wrong track (although "worst enemy" = inverse is right). Argh! Nobody's even close. I should add another hint. – Rand al'Thor Nov 19 '14 at 14:54

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.

• The cousin could be a number like .999... That turns into 1 at " the end of her life, " – Chris Cudmore Nov 18 '14 at 20:40
• That's what I mentioned - a floating point number. – Isaac Askew Nov 18 '14 at 20:48
• But this isn't rounding. At the end of its life (infinite digits) it actually equals 1 exactly. – Chris Cudmore Nov 18 '14 at 20:50
• This is a really nice answer (unfortunately I can't +1 for another 5 hours since I've reached my daily limit), but not what I intended. – Rand al'Thor Nov 18 '14 at 20:53
• I had the same answer. The little brother fits really fine, especially with multiplication. – njzk2 Nov 19 '14 at 19:06

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.

• Lovely idea! Not the intended answer though. (Is e^-x a sibling, as in your first spoilertagged bit, or a cousin?) You have the right interpretation of 'smooth', but it's only the younger brother and not the narrator who's smooth. – Rand al'Thor Nov 19 '14 at 10:47
• You're the closest yet in what you say about "worst enemy"! – Rand al'Thor Nov 19 '14 at 10:48

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?

• Another interesting idea, but nope. – Rand al'Thor Nov 18 '14 at 21:31
• This. This is ƒ^%#ing brilliant. – haneefmubarak Nov 19 '14 at 8:16
• Agreed, this is great, +1. – DivideByZero Nov 20 '14 at 5:18

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.

• Very good; you're the closest yet! Not quite there yet though. – Rand al'Thor Nov 19 '14 at 15:57
• @randal'thor Awesome. But this is all I know about this area of math, since I'm not an undergraduate yet. Guess someone else gotta take it from here now :D – nexolute Nov 19 '14 at 16:22
• @randal'thor Actually, upon thinking a little more, there is just one more thing I can do. Since we are talking about "who am I", I am just an element inside an equivalence class. Yeah? – nexolute Nov 19 '14 at 16:28
• No; you need to think on a broader scale. It's more on the level of "I am an equivalence relation" in the abstract (though that's not it). – Rand al'Thor Nov 19 '14 at 16:29

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.

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.

• This isn't what I'm thinking of, but if you have a good argument I'll use my last vote of the day to +1 it. Have you read the comments to the question? – Rand al'Thor Nov 18 '14 at 18:32
• Wave at my eldest brother still doesnt make sense – stackErr Nov 18 '14 at 18:33
• @randal'thor no, I didn't. – Raystafarian Nov 18 '14 at 18:34
• @stackErr stars are visible at different wave lengths, red and blue shift may cause them to appear to be at different stages. It's not perfect, but it is my guess. – Raystafarian Nov 18 '14 at 18:38
• Are you considering stars in an astrophysical sense (transforming into each other) or a geocentric sense (sun and moon)? And what about "We are the most perfect of our race, Because each of us has a clear worst enemy" and "he even steals my words, confusing all concerned"? It doesn't all fit together, but +1 anyway. – Rand al'Thor Nov 18 '14 at 18:39

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 $\frac{sin}{cos}$

• @stackerr already tried this - it's not the answer. But +1 (in 2 hours, when I can vote again!) for cousin <-> cosine; I love wordplay! – Rand al'Thor Nov 18 '14 at 21:55

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.

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

• Is subtraction a sibling or cousin of multiplication? And only four operations don't count as "many siblings". Sorry for sounding so negative; +1 anyway! I should probably add another hint soon, since everyone's barking up one of the same few wrong trees. – Rand al'Thor Nov 19 '14 at 10:19
• I liked this answer until the "brother, cousin, etc" part. Good thinking, though. – Raystafarian Nov 19 '14 at 12:41

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.

• The brothers and (especially) cousin bits still don't quite work, but you're (in some ways) getting closer! – Rand al'Thor Nov 19 '14 at 17:00

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.

• Another nice answer, but again the cousin and younger brother don't really work out. Again not abstract enough! – Rand al'Thor Nov 19 '14 at 16:41

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!

• I don't follow the logic for the brothers and cousin bit... – Rand al'Thor Nov 19 '14 at 17:00
• I'm thinking about the cousin in terms of a computational process, where the "end of her life" is the point where she has finished each step of the summation/sigma function? I am interpreting "turn into me" as producing an equivalent result to me. – Oscar Gillespie Nov 19 '14 at 17:16
• 'End of her life' is just a bit of wordplay... – Rand al'Thor Nov 19 '14 at 17:20

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

• +1, but neither of those is right. The second is still not abstract enough! – Rand al'Thor Nov 20 '14 at 20:10

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?

• Leipzig is a reference to a mathematician, but not the one you think! – Rand al'Thor Dec 3 '14 at 11:02