- after the 2014 mid-term elections

Tuesday was wet. We couldn’t have bothered to vote.
Panhandlers huddle like football players, anonymous,

under a sponsored freeway. Chubby constables
watch them from their warm cars. Politics

is what happens on FOX or CNN
in some distant country. We wait

in line for a hot latte to warm our hands
at Starbucks, before starting our second jobs

assembling complaints of failures on a phone line.


Copyright © 2014 Kurt Lovelace, All Rights Reserved

I started out believing in everything,
the open field, plow in hand, horse
waiting to be worked, words
hedged in the furrow, irises
open to the moment of opening, as if

posturing a proof were proof enough
but without the heavy lifting, of burdens,
the concrete blunders one must make, clearing
the way to ubiquitous insight. If only
our own desires would stop

helping the scrunched imp of all these days
rolled-up into aphasias of dreaming
that stream down like drops of sunlight
through the wet branches of Spring,
it might be enough. Perhaps

I can ask you about it, someday,
and you’ll tell me everything
I’ve ever wanted was within reach
if only I would have put out my hands,
wide palms like bells ringing, after they clap,

but as if at a wedding, wake, or just praise
at the hours and minutes granted us, I don’t know.

Put your fears in a little box
and smoke it, not like this warm interrogatory weather
we’ve been having, that peels
shirts from bodies with an utter unconcern that’s neither
here nor there.

It used to be beautiful but people got there
with ideas. I don’t know why a parking lot

should cover the green velvet moss that wrapped
the long slippery slate-stone path to the water

under thick green sun-spackled trees that was like walking
through golden pollen hovering inside the vest of a vast leprechaun

before opening out onto a beige pebbled beach
of bodies bobbing naked in the sunned shallows

or reclining like purposeful porpoises that Manet
or Seurat would gladly have painted, hips

and breasts, with their delicate French brushstrokes.
I decline the five dollar asking price

and drive on, back to Austin, talking to myself
feeling like Matthew McConaughey in a Mercedes commercial,

famous in my own mind, alone, and bewildered.


In Soonest Mended, through a humbling out of self by means of a sequence of self-referential questions and answers, through a process of bringing forth by rejecting and flipping answers and questions around, through a discourse that levels itself out by qualification as it proceeds, John Ashbery achieves a beautiful and stunning sublimation of self through the mere act of talking, thereby discovering, almost as if by accident, the nature of the poetic truth he had, apparently, been aiming for all along.

I posit that Ashbery hones a reductionist, almost mathematical, technique for approaching the truth. His speakers engage in a series of approximations to the “truth” using argumentative qualifications. It is precisely these qualifications that allow the speakers to wend their way to truth by discovering, recovering, and discarding the many “truths” — these weaker version of themselves — that argumentatively lie along the path of such self-referential discourse. This is a key technique used within almost all of Ashbery’s poetry: talk that uncovers truth by qualifying itself at every turn.

Ashbery takes his title from the second half of the proverb “Least said, soonest mended.” The “mended” here was originally “amended” and therein lies a piece of Ashbery humor, as Ashbery doubtless knows the history and origin of this proverb. And that Ashbery choose to leave out the first part of the proverb, namely “least said,” reinforces the notion that Ashbery does not want to say too little, but perhaps “just enough” or even saying “too much” — being exactly the opposite of the original proverb’s advice. It is this that is humorous in a typically cerebral Ashbery sense. So Ashbery chooses to place his focus on the proverb’s second part. Now, from “soonest amended”, we get the idea that one may wish to amend what one has just said by qualifying oneself, and perhaps the “sooner the better” — as implied by “soonest.” This is perhaps a peculiar idea, that we should amend what we say as soon as we say it, yet that is precisely what Ashbery does throughout this poem — constantly qualifying and refining what has been said, by digging around for the “true truth” of his speaker’s arguments rather than quickly settling on one as the unequivocal truth.

Ashbery begins the poem with the following eight lines:
Barely tolerated, living on the margin
In our technological society, we were always having to be rescued
On the brink of destruction, like heroines in Orlando Furioso
Before it was time to start all over again.
There would be thunder in the bushes, a rustling of coils,
And Angelica, in the Ingres painting, was considering
The colorful but small monster near her toe, as though wondering whether forgetting
The whole thing might not, in the end, be the only solution.

And, as if we had an empty Easter egg basket, we are off to hunt for Ashbery’s meanings — meanings that, at first reading appear to relish ambiguity over overt clarity. Even in the very first line, “Barely tolerated, living on the margin” we are led to wonder: what “margin” is the speaker referring to? The margin of a book? A beach? A writing table? Whatever the choice, margin implies an edge or boundary between two different states or an area of transition. Whatever the nature of this boundary, it is in an uncertain state of affairs.

This is now further clarified by the next line, “in our technological society” — so the speaker may be saying that they are non-technological or else otherwise disenfranchised from the larger “technological society” in some distinct manner. Ashbery is gay, and being gay during the 1950’s and 1960’s would certainly place oneself “on the margins” of mainstream society, especially during those years. Also, being a young writer, he would have been “on the margins” as a publishing poet and as a poet constantly writing new poems “in the margins.”

But as the poet goes on to say, “we were always having to be rescued / On the brink of destruction” — but rescued by whom? And how would one of these rescues proceed? Well, Ashbery’s answer is quite distinct: “like heroines in Orlando Furioso / Before it was time to start all over.” So, the speaker is always having to be rescued like a female heroine in The Frenzy of Orlando or, literally, Mad Orlando, an Italian romantic epic by Ludovico Ariosto that first appeared in print in 1516. However, Ashbery is more likely referring to the operatic interpretation of this epic by Vivaldi. It may be worth noting that Orlando Furioso was the continuation of Matteo Maria Boiardo’s unfinished romance Orlando Innamorato or “Orlando in Love.” And so too, Orlando Furioso is a love story. And it is filled with fantastic creatures such as the hipppogriff and a sea beast called an orc and many others. The story is episodic but shifts place rapidly, wondering all over the globe, from Europe to the Hebrides to Japan. At its core, it is the story of unrequited love between the christian knight Orlando and his pagan princess, Angelica. As the story unfolds, Orlando’s love quickly becomes obsession and devolves into madness. Eventually, Orlando’s friend and fellow knight, Astolfo, journeys to the moon using Elijah’s flaming chariot to bring back Orlando’s wits, because the moon is the place where everything lost on the earth may be found. Orlando, returned to sanity, falls out of love with Angelica, and resumes his duties as a good knight, helping his emperor, Charlemagne, kill King Agramante in battle.

Ashbery’s speaker identifies themselves with the heroine, Angelica, a feminine if not robustly homosexual reference if one assumes Ashbery to be referring to himself and fellow gay writers during the 1950’s. But this reference is immediately run into yet another reference to the story, almost as if it were a doppleganger reference to the first reference, and we are again given Orlando’s Angelica as imagined and painted by Ingres. These lines, with an almost toy-like serpent at Angelica’s feet, feel almost superfluous, except for the words they prompt the speaker to conclude these lines with, namely, “as though wondering whether forgetting / The whole thing might not, in the end, be the only solution.” Although we are not privy as to what “the whole thing” is that is to be, perhaps, forgotten, the finality of this conclusion resonates powerfully within these lines as one possible conclusion to the “barely tolerated, living on the edge” from the poem’s first line and “having to be rescued” from the poem’s preceding lines. Forgetting is a natural and powerful coping mechanism for dealing with loss, especially the loss of a love.

Now, from this point forward, the poem truly begins to mimic, albeit in a very scaled- down version, the wildly episodic shenanigans of Orlando Furioso — as next we meet “Happy Hooligan in his rusted green automobile” who “Came plowing down the course, just to make sure everything was O.K.” And the speaker continues with “Only by that time we were in another chapter and confused / About how to receive this latest piece of information.” Being already in another chapter is how we often find ourselves in life, even as we attempt to stand still to question what has just transpired — life continues onwards without regard around us and, so too, for Ashbery’s speaker.

This is the beginning of an astonishing process of qualification wherein the speaker explores other explanations, often contrary too what has just been said, yet immediately following a statement or pronouncement, as if intentionally negating the previous statement helps to get at the truth of things — a building up and breaking down process the continues its work throughout the poem.

Only now we were in another chapter and confused
About how to receive this latest piece of information.
Was this information? Weren’t we rather acting this out
For someone else’s benefit, thoughts in a mind
With room enough and to spare for our little problems (so they begin to seem)
        Our daily quandary about food and the rent and bills to be paid?


Here we witness a marvelous shifting away from one explanation by suggesting a possibly superior alternative explanation: “Was this information? Weren’t we rather acting this out.” This brings up the question of what information itself is, and in the context of a poem, what information are we receiving? Also, “piece of information” implies that it is always incomplete, part of something before itself and after itself, a mere chunk in the middle of a stream of data that is coming at us from the world. But the “weren’t we acting this out / For someone else’s benefit, thoughts in a mind / With room enough and to spare “ aside from a stage with actors performing in front of a audience — something a poem always is, in a sense, doing — also implies social norms and doing what is expected of us but suffused with even a hint or allusion to God, by virtue of the “ampleness” of the mind that can encompass “our little problems” — and all of these suggestive meanings are wrapped up in a quick marvelous short catalog that Ashbery here jots out as “Our daily quandary about food and the rent and bills to be paid?” that concludes this sentence.

Ashbery’s speaker then swings away at yet another possible solution, with:

To reduce all this to a small variant
To step free at last, minuscule on the gigantic plateau –
This was our ambition: to be small and clear and free.

But immediately realizes that this would be impossible, and so quickly corrects with:

Alas, the summer’s energy wanes quickly,
A moment and it is gone. And no longer
May we make the necessary arrangements, simple as they are.

But of course. the arrangements that solved all of this, the reduced it all to “a small variant,” were never simple. Then comes an enigmatic statement followed by a corrective injection of reality, as if in the same breath:

Our star was brighter perhaps when it had water in it.
Now there is no question even of that, but only
Of holding on to the hard earth so as not to get thrown off.

Water may act as a magnifying glass or as a mirror reflecting stars at night in a pool of water, so their star would be brighter when it had water in it. But then, this getting thrown off the earth is tempered immediately by the softness of a vision:

With an occasional dream, a vision: a robin flies across
The upper corner of the window, you brush your hair away
And cannot quite see, or a wound will flash
Against the sweet faces of the others, something like:
This is what you wanted to hear, so why
Did you think of listening to something else?

Indeed, why would we think of listening to something else if we are listening to what we wanted to hear? This is one of the most common situations in life, getting what we think we want only to ponder something else, the thing we do not have as if that which is not present has more power over our thoughts than what is in front of us. This is often the case, and Ashbery is right in stating it clearly. And he continues to mine truth along this rich vein that he has here struck, that conclude Ashbery’s first stanza:

This is what you wanted to hear, so why
Did you think of listening to something else?
We are talkers It is true, but underneath the talk lies
The moving and not wanting to be moved, the loose
Meaning, untidy and simple like a threshing floor.

And indeed, we are all talkers with hidden agendas, and often waiting for others to move us while ready to fight such movement within ourselves. And often, meaning can be “loose” in the sense that we are just “chatting” to no real purpose other than talk for the sake of social cement, but Ashbery captures this with the superb image of “loose / meaning, untidy and simple like a threshing floor” where threshing floor is the flat surface a farmer uses to thresh or winnow grain from the harvest. This is perhaps an oblique yet lovely reference to John Keat’s To Autumn, Stanza 2, lines 2-4:

Sometimes whoever seeks abroad may find
Thee sitting careless on a granary floor,
Thy hair soft-lifted by the winnowing wind:

As we enter the second and final stanza of this poem, the speaker continues with:

These then were some hazards of the course,
Yet thought we knew the course was hazards and nothing else

This then being a statement with an immediate acknowledgment of it being too true or not truthful enough as the “course” does not merely contain hazards on it, but it is composed entirely of hazards, from beginning to end — not unlike life itself. Then we rapidly move to:


It was still a shock when, almost a quarter of a century later,
The clarity of the rules dawned on you for the first time.

as time shifts forward 25 years and revelation hits the speaker, that their understanding of “games” was backwards, and that:

They were the players, and we who had struggled at the game
Were merely spectators, though subject to its vicissitudes
And moving with it out of the tearful stadium, borne on shoulders, at last.


So it is the entire “game” that is being hoisted onto the shoulders and moved out of the “tearful stadium” and “at last” as if the time had not come soon enough. The key word here is “spectators” form the Latin spectare, ‘to gaze at, observe’ , which is related to ‘spectacle’ — again, what a poem does, in a sense, is perform a public spectacle. But Ashbery here shifts the “players” for the “spectators” around, and has the stadium carry the players out, and odd mixed metaphor. And next we see these lines:

Night after night this message returns, repeated
In the flickering bulbs of the sky raised past us, taken away from us
Yet ours over and over until the end that is past truth,
The being of our sentences, in the climate that fostered them,
Not ours to own, like a book, but to be with, and sometimes
To be without, alone and desperate.

Here the “message” that we are not the players but the spectators is “repeated / In the flickering bulbs of the sky” but these are “raised past us, taken away from us” but somehow “ours over and over until the end that is past truth” and this might certainly be death, which is an “end that is past truth.” Here, Ashbery makes a subtle yet overt allusion to Wallace Stevens, for so loaded is the word “climate” next to the word “sentences” that no one can escape the shadow of Steven’s previous usage of those words. And so too this truth is “not ours to own, like a book” but interestingly to “be without, alone and desperate.” Then the speakers move onward, realizing that:

But the fantasy makes it ours, a kind of fend-sitting
Raised to the level of an esthetic ideal.

As indeed a fantasy will make anything ours, but only for the duration of the fantasizing. And academics have naturally made this into a “kind of fence-sitting” where we take all sides at once, and thereby make this our esthetic ideal. And Ashbery continues his catalog with:

These were moments, years,
Solid with reality, faces, namable events, kisses, heroic acts,
But like the friendly beginning of a geometrical progression
Not too reassuring, as though meaning could be cast aside some day
When it had been outgrown.

Which, of course, is absurd, as meaning can never be simply “caste aside” nor outgrown. But it is an enjoyable fantasy for the speaker. But then the speaker adds yet another “correction” or qualification with:

Better, you said, to stay cowering
Like this in the early lessons, since the promise of learning Is a delusion,

So learning is a delusion, is yet another “answer” for the speaker who then qualifies this yet again:

and I agreed, adding that
Tomorrow would alter the sense of what had already been learned,
That the learning process is extended in this way, so that from this standpoint
None of us ever graduates from college,
For time is an emulsion, and probably thinking not to grow up
Is the brightest kind of maturity for us, right now at any rate.

But there is inescapable humor and truth in the idea that “none of us every graduates.” Now, that “time is an emulsion” is a lovely image coming from the technical world of photographic film development and chemistry, This returns us to the “margin” where the concept of a boundary or mixing place between two different states of matter occurs naturally comes to us again, for emulsions are part of a class of two-phase systems where one liquid is mixed into another. Emulsions do not have a static structure but rather are best described using statistics — the mathematics of pinning down what generally cannot be pinned down, and so essentially an approximation that yields useful albeit limited information.

But the metaphor of time being an emulsion is fascinating and highly novel, to say the least. Yet the speaker qualifies even this wonderful metaphor with “probably thinking not to grow up / Is the brightest kind of maturity for us” but only “right now at any rate” — as if in the next minute the speaker might decide otherwise, almost exactly like molecules mixing and changing second by second in the emulsion of time that we find ourselves.

And time is indeed uncertain , for we now come into the poems final poetic machinations and stances. At first, the speaker decides upon generosity by offering up:

And you see, both of us were right, though nothing
Has somehow come to nothing;

But of course, even conceding that both or all arguments were “right” means little if everything has “somehow come to nothing” — an all negating conclusion — which make the concession appear less generous than at first it might have seemed. And the speaker here continues:

the avatars
Of our conforming to the rules and living
Around the home have made—well, in a sense, “good citizens” of us,
Brushing the teeth and all that, and learning to accept
The charity of the hard moments as they are doled out,
For this is action, this not being sure, this careless
Preparing, sowing the seeds crooked in the furrow,
Making ready to forget, and always coming back
To the mooring of starting out, that day so long ago.

The word “home” here is highly charged as it describes much more than merely living in a house, but our entire circumstances of living in our “technological society” were we do get to “brushing the teeth and all that” but along with everything else that we do in our lives. In a sense, everything in life happens “around the house.” But now Ashbery turns fully into the poetic sublime, “learning to accept / The charity of the hard moments” for they are indeed a kind of charity, the moments where all of our preparations come together, and we must finally act and choose and do rather than think and describe. But herein the speaker fully curls up into the nature of time by embracing it “for this action, this not being sure, this careless preparing, sowing the seeds crooked in the furrow” is but “making ready to forget.” Yet the speaker is not finished as they realize that they are “always coming back / To the mooring of starting out, that day so long ago” of their poetic beginnings as to the beginning of their life, and you have the implied yet almost unexpected cliche “life is like a boat ride” left dangling in midair as the poem ends, but the true meanings brought forth lie somewhere far off and reverberate beyond the page.

So, having now worked through the entire text, line by line, and pointing out each internal turn or qualification by the speaker, it becomes apparent that Ashbery has gotten a great deal of distance and motion out of this technique of talking things out by extending talk through constant acts of qualification — ending up with talk that turns upon itself, that decides that there are better answers to be had further down in the discussion. And it is a profound millage. Beginning with “barely tolerated” and arriving at “to the mooring of starting out, that day so long ago,” Ashbery has brought us full circle and back to the beginning by a circling suffused with a delightful new understanding of our limitations and of our inventive capabilities within the confines of language.

Poetry cannot be translated. So too, other than for giving a reader the raw idea of what is being said, any literal translation is especially negligible. The translator must be someone capable of writing decent if not inspired English poetry, whatever that might mean. Therefore, if one does attempt the translation of a poem, it must be ruthless, a culling from the bloody heart of both tongues. In other words, one may only create a poem in the target language that is its own poem with echoes, distortions, and intentions pointing to those places that the original also points out. So, that being said, here are three of my favorite poems from the Spanish, German, and French by Neruda, Rilke and Prevert, respectively.

These translations are very much, in an effort of love and intellect, an attempt to convey the beauty, wordplay and sound-play in the originals — and this explains some of my  perhaps more daring  “choices” — first person tense, ellipses, metaphor shifts, occasional sound emphasis over word sense — choices I prefer to call “intelligently risky”  as they are collectively my attempts to “transmogrify” these lovely poems into some semblance of a worthy English simulacrum.  Caveat emptor!

Anoche al apagar la luz
se me durmieron las raíces
y se me quedaron los ojos
enredados entre las hojas
hasta que, tarde, con la sombra
se me cayó una rama al sueño
y por el tronco me subió
la fría noche de cristal
como una iguana transparente.
Entonces me quedé dormido.
Cerré los ojos y las hojas.
Pablo Neruda


Last night, putting the light
out, my deep roots slept
but my eyes strayed, open
tangled in between leaves
until, later, the shadow
of a branch fell over my dream
and I rose up into the trunk
of the cold night, a crystal
transparent iguana.

I slept soundly then.

I closed my eyes and my leaves.

translated by Kurt Lovelace,
Copyright 2014 All Rights Reserved


Herr: Es ist Zeit. Der Sommer war sehr groß.
Leg deinen Schatten auf die Sonnenuhren,
und auf den Fluren laß die Winde los.

Befiehl den letzten Früchten voll zu sein;
gieb ihnen noch zwei südlichere Tage,
dränge sie zur Vollendung hin und jage
die letzte Süße in den schweren Wein.

Wer jetzt kein Haus hat, baut sich keines mehr.
Wer jetzt allein ist, wird es lange bleiben,
wird wachen, lesen, lange Briefe schreiben
und wird in den Aleen hin und her
unruhig wandern, wenn die Blätter treiben

Rainer Maria Rilke

Fall Day

God! Its about time! Summer was so fat.
Now shadows drag over the sundials,
and in meadows the wind rips loose!

Oh, let our last fruits fatten into fullness;
give us just two ripe sun-drenched days
plumping all into ripeness till hither and thither
the last sweet drops drain into swarthy wine.

Whoever has no house, you will not build it now.
If you are alone, it will be for a long time,
you will stay awake, reading, writing long letters
as you walk alone, shuffling, here and there,
disturbed, wandering where the leaves tremble.

translated by Kurt Lovelace,
Copyright 2014 All Rights Reserved

Déjeuner du matin

Il a mis le café
Dans la tasse
Il a mis le lait
Dans la tasse de café
Il a mis le sucre
Dans le café au lait
Avec la petite cuiller
Il a tourné
Il a bu le café au lait
Et il a reposé la tasse
Sans me parler
Il a allumé
Une cigarette
Il a fait des ronds
Avec la fumée
Il a mis les cendres
Dans le cendrier
Sans me parler
Sans me regarder
Il s’est levé
Il a mis
Son chapeau sur sa tête
Il a mis
Son manteau de pluie
Parce qu’il pleuvait
Et il est parti
Sous la pluie
Sans une parole
Sans me regarder
Et moi j’ai pris
Ma tête dans ma main
Et j’ai pleuré.

Jacques Prevert

Breakfast at the Dinner

He pours coffee
into the cup.
He pours milk
into the cup of coffee.
He sprinkles sugar
atop the café au lait,
and with a little spoon
stirs it round.
He finishes his café au lait.
He reposes cup in saucer.
Not one word spoken,
he lights up
a cigarette.
He blows round
smoke rings.
He taps ash
into the ashtray.
Never speaking to me,
never regarding me,
he readies to leave, places
hat on head,
throws on his coat
because rain is splashing down
pouring into puddles
as he leaves me,
turning into the splashing
rain, never speaking
nor regarding me once.
And I, eyes
splashing into the ground
of my hands,

translation by Kurt Lovelace
copyright 2014 All Rights Reserved

Note: this article is intended for undergraduate math majors, preferably seniors, but freshman will benefit from the look ahead, at what is expected to be known by them when they are seniors, as well as entering graduate students.

I am readying myself for the GRE math subject exam in the late Spring, and therefore reviewing all four years of undergraduate mathematics at this time.

In what follows, I will be summarizing every major area of undergraduate mathematics, as follows:

  1. Geometry: Plane, Elliptic, Hyperbolic, and Affine
  2. Linear Algebra
  3. Vector Analysis
  4. Real Analysis
  5. Complex Analysis
  6. Topology
  7. Ordinary Differential Equations
  8. Fourier Analysis, Lebesgue Integration and General Transform Theory
  9. Probability Theory
  10. Abstract Algebra
  11. Graph Theory
  12. Combinatorics and Algorithmic Complexity
  13. Set Theory and Transfinite Arithmetic
  14. Basic & Analytic Number Theory
  15. Partial Differential Equations
  16. Differential Geometry

If you are a student of any subject, then I hope you have already asked yourself at times the question “What does all of this mean? What is it for?” if you are as I am, a mathematics major, then I believe that this question is particularly important. For one can too often get lost in the forest, standing amid the numberless trees, each one a bit different from its neighbors yet all oddly familiar, somehow similar, like wondering through a waking dream.

This is neither an idle nor a superfluous question. Call it “the big picture.” Call it what you will. It is important to know the gist and the connections between different areas of ones subject, and to know each area for what it really is about.

Linear Algebra

Let’s look at linear algebra. The matrix is the lingua franca tool of linear algebra, and so linear algebra is the study of vector spaces and their transformations using matrices. After a first course, one should know the axioms that define a vector space, the algebra of matrices, how to express the structure of vector spaces using matrices, and especially how, given the basis of a vector space, to represent any transformation of that vector space using a matrix, and how to use and calculate eigenvalues and eigenvectors. Lastly, the key theorem of linear algebra is to know the 8 distinct conditions that each alone can guarantee a matrix is inevitable. And that is it, the heart of an entire semester of undergraduate linear algebra summarized in one paragraph. Again, in a nutshell: basic linear algebra is the study of the transformation of vector spaces into one another and understanding equivalent conditions under which a matrix is invertible.

What is linear algebra used for? Everything, and almost everywhere, is the short answer. Anytime you seek a first approximation to some problem, it is likely that you will be using linear algebra. In graph theory matrices are used to represent a graph’s structure in a precise and concise manner. In abstract algebra, matrices are used in the representation of groups and other algebraic structures. And linear algebra comes up in both ordinary and partial differential equations, differential geometry, knot theory, number theory, and almost everywhere else. Learn it well, know it well if you are going to be doing research in pure or applied mathematics.

If one takes a fourth year, second semester of linear algebra, often considered “part 2″ of linear algebra studies, then one will likely encounter — inner product spaces, direct sums of subspaces, primary decomposition, reduction to triangular and Jordan forms, both rational and classical forms, dual spaces, orthogonal direct sums, bilinear and quadratic forms, and real normality — among the main topics.

Here are some great free full textbooks to download for reference or study:

Linear Algebra by Jim Hefferon. 499 pages. [direct link to whole book.]

Linear Algebra. David Cherney, Tom Denton & Andrew Waldron. 410 pages. [direct link to whole book]

-=KuRt=- © 2013 Kurt Lovelace – All Rights Reserved

Euler’s Characteristic Formula V-E+F = 2

How is it that for thousands of years the best minds in mathematics did not see the fundamental relationship that, in any regular polyhedron, the sum of the vertices and faces minus the edges equals two? Although this question is interesting, no attempt will be made to answer it. Yet the question, merely by being asked, serves to highlight the tremendous stature of Leonard Euler.

Euler worked from first principles, digging into a topic by performing calculations to get a feel for the shape and edges of a problem, then developed conjectures and proofs based on such research. And so, Euler must have tabulated the edges, faces and vertices of shapes such as the platonic solids to thereby notice this relationship. Perhaps it took an almost childlike playfulness to discover this relationship. Yet it is all speculation. The only fact is that Euler did discover that V-E+F=2 where V, E, and F are, respectively, the vertices, edges and faces of a regular polyhedron

I first learned of Euler’s formula in a senior course on graph theory taught by the Polish graph theorist Dr. Siemion Fajtlowicz. Therefore, let me provide a few definitions before offering a compact proof that V-E+F=2 using basic graph theoretical methods.

Definition. A graph consists of a non-empty set of vertices and a set of edges, possibly empty. If an edge E exists, then it will be related to an unordered pair (a,b) of vertices in V. We may write G=(\{V\},\{E\}) to denote the graph.

A graph is finite if both the vertex set and edge set are finite. A graph that is not finite is infinite. The size of a graph is the number of vertices. The order of a graph is the number of edges. All graphs discussed in this article are finite.

Immediately note that if we are given one edge, then there exist two vertices — because an edge must connect to something and that something will always be a vertex. Two vertices are incident if they share a common edge and said to be adjacent. An edge with identical ends is called a loop while an edge with distinct ends is called a link. A graph is simple if it has no loops.

The degree d(v) of a vertex in a graph G is equal to the number of edges in G incident with v where each loop counts as two edges. Thus, a vertex with two edges has degree 2 because the degree of the other two vertices would be one each, and so the sum of the degrees of all the vertices would be 4. From this we have immediately the following two theorems.

Theorem 1. The sum of the degree of the vertices of a graph equals twice the number of edges. That is

\displaystyle \sum\limits_{i/G}{d({{v}_{i}})}=2e

proof. If a graph has no edges, then e=0 and the theorem is true because the sum of the degree of the vertices of a graph with no edges must, by definition, be zero. If a given vertex has an edge, then there must be a second vertex connected to the first vertex by the given edge and so the sum of the degree of the vertexes will be 2, even if the edge were a loop beginning and returning to a single vertex the degree of that vertex due to the loop would still be 2.

Now, each additional edge added to the graph will increase either the degree of the vertex it is connect to and add one additional vertex or else it will add one degree each to two vertices that already exist. Thus, each time an edge is added, the degree count of the vertices increases by 2. Therefore, the sum of the degree of all the vertices will always be simply twice the number of edges present.

Theorem 1.1 The number of vertices in a graph of odd degree is even.

proof. Let {{v}_{1}},{{v}_{2}} be sets of vertices where the {{v}_{1}} is of odd degree and {{v}_{2}} is of even degree. Then, from Theorem 1, we must have that


but we have already established that


and therefore we must have







is even. Thus, we have that the sum of the vertices in a graph of odd degree is even. Now, we will need the following definitions to proceed.

Definition. A cycle is a closed chain of edges. A connected graph that contains no cycles is a tree.

Definition. A spanning tree of a graph G is one that uses every vertex of G but not all of the edges of G.

Every connected graph G contains a spanning tree T as a subgraph of G.

Definition. A planer graph is one that can be drawn in the plane without crossing any edges.

Definition. A face in the plane consists of both unbounded and bounded regions.

The last definition allows us to say that the number of “faces” of a finite line segment in the plane is 1, but that the number of faces of an infinite line is 2, one for each side that the line divides the plane into, and that the number of faces of a circle in the plane is 2, one corresponding to the inside and the other to the outside of the circle one being the unbounded and the other the bounded region.

Now we are ready to prove Euler’s formula as it may be stated in graph theory.

Theorem 2. If G is a connected planer graph with vertices v, edges e, and faces f, then

\displaystyle v-e+f=2

proof. Let T be a spanning tree of a graph G. Then the number of faces f=1 and the number of edges e= v – 1 is true for the spanning tree T of G and so we have v-(v+1)+(1)=2 and our formula is true. Now, G may be constructed from T by adding edges to T, but each time we do so we are adding a new face to T. Therefore, with the addition of each new edge, e will increase by 1 and f will increase by 1. Therefore, all of these additions cancel out and we have our formula.

Now, a homomorphism exists between polyhedron and graphs in that a connected plane graph can be uniquely associated with a polyhedron by making any face the flat unbounded part of the plane. Therefore, Euler’s formula is true for polyhedron. We can therefore immediately prove:

Theorem 3. Exactly five platonic solids exist.

proof. Let the number of edges and faces be n and the number of edges each vertex is incident to be d. If we multiply nf, we get twice as many edges because each edge belongs to two faces, so nf is the number of faces multiplied by the number of edges on each face. Likewise, with dv each edge is incident to two faces so that we have dv=2e so that we have the equality



e=\frac{dv}{2} and f=\frac{dv}{n}

Now, placing these into Euler’s formula, we get




But both v and n are positive integers and for (2n+2d-dn)v=4n to be true, we must have that




and so there are just five possibilities for the values of n and d and each of these corresponds to one of the five platonic solids, so that we have

n=3,d=3 is a tetrahedron
n=3,d=4 is a hexahedron also known as a cube
n=4,d=3 is a octahedron
n=3,d=5 is a dodecahedron
n=5,d=3 is a icosahedron

the 5 Platonic Solids


© 2012 Kurt Lovelace – All Rights Reserved

At Marfreley’s Bar in Houston, Texas

In a dim lit mural behind the bar,
two swans amble in front of a plantation:
its white house lies against the river, lonely

for the cover of more trees that the artist
left out, as the rushing water
empties into the dark dandelion breeze

of rewritten histories. And I had wanted to see
a single woman out, tonight, sitting
alone, like me at the bar, looking

at their life, the plantation, the swans swallowing
small sips of whatever they find in front
of themselves, any parts of a life that might

make sense, tell me I have done the right things.

© 2012 Kurt Lovelace – All Rights Reserved


Ghazal to Disquietude


Drowned in the honk-squeal above the guard rail, I can almost hear
waves sweep in as the soft susurration in the tip of their lips melts the sand between my curled toes.

Noise is now everywhere I want to be
without it. Cars swoosh past Galveston beach roaring their inept monstrous lungs. I can barely

breathe. Or think. Why do trees and blades of every green thing shudder?
Because we are a hyper-intelligent insidious poison? Cats and dogs cling to us in shock and awe.

Ninety-five percent of a car’s energy goes towards moving simply itself not the passengers.
Or rather that’s 2,500 pounds of wastefulness before the crux of tissue steering the steel.

In Hermann Memorial Park a yellow-blue finch tries to sing and fails
in the roar and wall of sound the cars shed in their wake on the I-10 adjoining the beige greenery.

I nod off under a canker tree. A whale whistles out of its water fountain, breathing.
I roll under such plushness, floating with barnacles and sticky ambergris. So glued are our dream’s illogical logic.

I am a sticky carbuncle tearing through the earth’s thin breathability. It’s afternoon in Houston.
I shower again. I scrunch into a starched shirt. I rope my throat with a dead worm’s shiny excrement.


My right ear is dead. When I was three
German measles like dappled freckles grew in me

killing the nerve. Now, left ear still good, I hear pretty well
the unprettiness in my parents voices as they divorce:

the light fades as I listen in, on the mosquito bitten dark
roof above the living room window, then roll on my back

to swallow my insignificance in the drifting milky way above.
Now the frogs have started up. A few ducks quack. A splash

might be catfish come to nibble at the stars
tangled in cheap tabloid floating on the pond’s scum.

Pain makes a squelch in my chest like tossed gravel
settling into decay layer at ponds pitch black bottom.

© 2012 Kurt Lovelace – All Rights Reserved


Grading the Weekend

While sipping coffee, I read what one student wrote:
“The surviving fifty rare whooping cranes
with their seven-foot wingspread that propels them
in their annual migration from northern Canada
to the Gulf of Mexico fly unerringly and
swiftly overhead as they migrate southward
using a kind of built-in radar
in their search for winter quarters
near Aransas Pass.”

Surviving fifty myself, feeling rare and whooping
with my six-foot slouch that propels me nowhere
in my daily migrations from the kitchen to the couch,
I live by the Gulf of Mexico, sleep unerringly and
swiftly, undercover, my dreams migrate southward
using a kind of built-in slinky
in search for vaginal quarters
near my wife’s Aransas Pass.

To be surviving melanoma is rare
with its seven wretched drugs I puke, that propels
me out of the gothic hospital to monthly migrations of chemo;
swimming in the Gulf of Mexico, on my back, I float unerringly and
slowly, overheard, the nurses’ whispers migrate southward
out of memory, which is a kind of built-in shit-breeder
when I am in pain and searching for the way out
near the dark rings of Uranus.

But survival is everything rare as whooping
or her pubic hair spread to propel me
in my daily migrations from her coffer to wherever
it is in the Gulf of Mexico I am off to, I unerringly
admit to caring enough to love her butt
less than I ought too as I migrate southward
using a kind of built-in stupidity
in my blindly succumbing to what is expected of me
clearly perfecting it into a fairly fucked life.

© 2012 Kurt Lovelace – All Rights Reserved



See the purple and green crayon alphabet scrawled on yellow sticky notes stapled to tiny Glen Hills cardboard orange juice containers sucked empty by a strawberry-headed freckled girl named Melissa Alexander Winsum,

See the cardboard, folded and wax coated, that once held the orange juice within it, was wood that came from somewhere green and quiet with squirrels that stretched out on the upholding limbs sucking towards the sun their green certitude of elm or pine or oak,

See how Melissa tied together her carton creation with thick pink fuzzy wool string pulled through holes in the juice containers pricked with a three-fifths whittled down number two Venus pencil she over sharpened while working excited in Miss Thurstin’s after school art class last Tuesday,

See how the wool string grew out of a sheep’s skin, that then kept it warm through a snowy Spring, how that wool sprouted, cell upon cell, a protein made from the very grass the sheep was grazing on, from x-ray sun to chlorophyll to sheep’s cud chewing transformed to the wiry gray mat of wool dyed pink, now holding aloft 26 spent juice containers wobbling in the wind the whole of our English alphabet.

© 2012 Kurt Lovelace – All Rights Reserved


Midnight Recital

Kneeling to untangle my dog’s leg from its leash,
how did I get here, walking a pit bull in the dark
under the sour leaves of drought resistant Texas oaks?
How have these years colluded to put me
with a woman who doesn’t like to be touched
as if my life were still attached
to a former life, lived in felt robes, kneeling,
questioningly, before God’s dead silence?
Why do I sometimes whisper beatitudes in Latin
when grinding roasted coffee beans for breakfast?
Why can’t a fuck be just a fuck like breathing
or the necessary forward movement of starlight
entering my eyes from Polaris when I look up?

Why is my life so intertwined that it folds me
into fractal compartments that expand, as if
from each decision, outward, new enclosures grip me
as I venture forward, faster than any logic I can conjure?
Should I kill politicians to address society’s wrongs?
Or open a shop and sell cracked imported Chinese
Chia Pets? Or get to the lunar surface to erase
the names of loved ones astronauts left behind?
How can this sticky motion of salt and water
hoisted on these dry branches of bone
discern a purpose, lost among thin pricks of starlight
that amble like ancient animals into the night?

© 2012 Kurt Lovelace – All Rights Reserved


Put Some Relish on Your Plate, Pontius Pilate

I started out believing in everything:
the open field, plow in hand, horse
waiting to be worked, words
hedged in the furrow, irises open
to the moment of opening

as if posturing a proof were proof enough
by itself, but without the heavy lifting of burdens,
the concrete blunders one must make,
clearing the way
to ubiquitous insight. Now, if only

my own desires would stop helping the scrunched imp
of all my days, rolled into aphasias of dreaming,
that stream down like sunlight
through the wet branches of Spring,
it might be enough. Perhaps

I can ask you about it someday
and you’ll tell me everything
I’ve ever wanted was within reach
if only I would have put out my hands,
wide palms like bells ringing

but as if to a wedding, a funeral
or just praise
at the hours and minutes granted to us,
I don’t know.
Say again, “What?”

Put your fears in a little box and smoke it
not this warm interrogatory weather
we’ve been having, that peels
shirts from bodies with an utter unconcern that’s neither
here nor there.

-=KuRt=- © 2012 Kurt Lovelace – All Rights Reserved



I grasp the impulse that might be driving you
to pity me in some odd way for being flabby and fifty
to your skinny and twenty, but you know, I like most
people stopped aging in my head at twenty-one, the
mental self-image of a nonstop Sid vicious, smiling at
you still trying to figure yourselves out, while we
older folk are done with nothing and wondering
everywhere we still can, asking better questions than
the thin shit we dredged up in our well-spent
grassy laid bare-assed whistling halleluiah youth.
And you listen to nothing we say all day with piercing
eyes as we watch you climbing our mistakes.

© 2012 Kurt Lovelace – All Rights Reserved

This week, Professor Siemion Fajtlowicz assigned two home work problems:

1. How many graphs with vertices 1 … n are there?

2. Up to isomorphism, how many graphs are there with n vertices?

3. If you invite 6 random people to a party, show that 3 of them will know each other or 3 of them will be mutual strangers — and show that this is guaranteed to always be the case — but only if you invite a minimum of 6 random people to the party. It will not be the case if we only invite 5 random people or 4 random people, et cetera.

Question 1 and 2 may appear to be identical questions, but they are not at all identical though they are related.

Question 1 is asking for the number of elements of S. Question 2 is asking for the number of elements of the quotient G\S, a very different and much more difficult question.

Theorem. There are

\displaystyle {{2}^{\left( \begin{matrix} n \\ 2 \\ \end{matrix} \right)}}

graphs with vertex set \displaystyle \{1,2,3,...,n-1,n\} .

Proof. The question is easily answered. Given n vertexes, if we start from vertex 1 and connect an edge to each of the remaining vertexes, so that 1 goes to 2, 1 goes to 3, and so on until 1 goes to (n-1) , and 1 goes finally to n. Now, including the null edge or simply 1 itself connected to nothing, which is a legitimate graph, we have that there are n graphs. Repeating this for each additional vertex, and taking into account that the edge may originate at either one of the vertexes making for two graphs — if allowing full duplicates — then we will have that there are exactly,

\displaystyle ^{1+2+3+...+(n-1)+n}

possible graphs raised to the power of 2 to account for starting at either vertex, which gives us:

\displaystyle {{2}^{1+2+3+...+(n-1)+n}}={{2}^{\sum\limits_{i/{{\mathbb{Z}}_{+}}}{i}}}={{2}^{\frac{n(n+1)}{2}}}

Therfore, the number of graphs with vertices 1 to n are:

\displaystyle {{2}^{\left( \begin{matrix} n \\ 2 \\ \end{matrix} \right)}}

and this is simply a more compact way of writing the RHS of the previous result using binomial coefficients. QED


Theorem. Up to isomorphism, there are

\displaystyle \frac{1}{n!}\sum\limits_{n\in \sum\nolimits_{n}{{}}}{{{2}^{o\left( \sigma \right)}}}

graphs with n vertices.

The answer to the question is much more difficult, because the answer involves some hefty but basic machinery from Group Theory and Combinatorics involving Burnside’s Lemma and Polya’s Enumeration Theorem such that this question may be reworded in those terms, so that it becomes:

How large is the set \displaystyle \sum\nolimits_{n}{{}}\backslash S where \displaystyle \sum\nolimits_{n}{{}} denotes the symmetric group on n letters and \displaystyle S is the set of graphs with vertex set \displaystyle {1,2,3,...,n-1,n} ?

The answer is

\displaystyle \frac{1}{n!}\sum\limits_{n\in \sum\nolimits_{n}{{}}}{{{2}^{o\left( \sigma \right)}}}

where \displaystyle o\left( \sigma \right) denotes the number of \displaystyle \sigma orbits on the set.


At any party with 6 guests, either 3 are mutual friends or else 3 are mutual strangers. That is, the symmetric Ramsey Number R(3,3) = 6

If we consider just one person, then the other five must fall into one of the two classes of being either a friend or a stranger. This follows immediately from the pigeonhole principle, namely that, if m pigeons occupy n holes where \displaystyle n<m , then at least one hole contains:

\displaystyle \left\lfloor \frac{m-1}{n} \right\rfloor +1

pigeons, where \displaystyle \left\lfloor {} \right\rfloor is the greatest integer function. The proof of this follows from the fact that the largest multiple of n that divides into m is the fractional part left over after n divides m-1 or

\displaystyle \left\lfloor \frac{m-1}{n} \right\rfloor

and so for n pigeons, we get

\displaystyle n\left\lfloor \frac{m-1}{n} \right\rfloor

pigeons that could be put into \displaystyle \left\lfloor \frac{m-1}{n} \right\rfloor holes. But we have m pigeons, and so there must be more than this many pigeons in the holes.

Now, for our problem, we have two classes, friends and strangers. If we choose one person, then that leaves 5 people with whom that person is either a friend or a stranger. And so, by the pigeonhole principle, for five objects going into two classes we must have at least:

\displaystyle \left\lfloor \frac{5-1}{2} \right\rfloor +1=3

members in one of the classes. In terms of a graph theoretical viewpoint, we have, starting with one vertex extended out to the other five “people” vertexes, that:

where the red edges represent friends and the blue edges represent strangers. We easily see that, regardless of how we choose one of our 2 colors, 3 of them must be blue or else 3 of them must be red, for if we have five red, then we have met the required condition that at least 3 be red, and likewise if 4 are red, we again fulfill the requirement that at least 3 be red; and so, this whole sequence of argument applies if we swap blur for red. Therefore, we are always guaranteed that there are 3 mutual friends or else that there are 3 mutual strangers in any group of 6 people. QED

In terms of Ramsey Numbers, the statement would be written \displaystyle R(3,3)=6 .

This is utterly fascinating, because what this is really telling us it that there is a type of structure built into any random finite set. In this case, for any binary operation or else any question or property that has two values, we have it that any finite set of 6 is sufficient to support there being 3 of one or 3 of the other of that property, and that one of the two sets of 3 always exists inside of the 6 items.

This is a glimpse into a type of “deep structure” embedded within the fabric of finite sets. This is more than merely surprising, as one should not really be expecting to find any such structure whatsoever in a random set.

Going somewhat beyond this homework problem, from experimental math studies using Mathematica, I conjecture that:

\displaystyle R(n,n)=\frac{1}{12}(3{{n}^{4}}-20{{n}^{3}}+63{{n}^{2}}-82n+48)

holds for n<=20 in a completely trivial and weak algebraic sense that the Ramsey Numbers in this range do indeed appear in this formula.

However, there is no natural or reasonable theoretical connection whatsoever between this formula and Ramsey Numbers other than the search for generating functions and sequence matching studies that I conducted. So, for n = 3 to 20 this formula yields:

\displaystyle \text{R(3,3) = 6}
\displaystyle \text{R(4,4) = 18}
\displaystyle \text{R(5,5) = }49
\displaystyle \text{R(6,6) = }116
\displaystyle \text{R(7,7) = }242
\displaystyle \text{R(8,8) = }456
\displaystyle \text{R(9,9) = }793
\displaystyle \text{R(10,10) = }1,294
\displaystyle \text{R(11,11) = }2,006
\displaystyle \text{R(12,12) = }2,982
\displaystyle \text{R(13,13) = }4,281
\displaystyle \text{R(14,14) = }5,968
\displaystyle \text{R(15,15) = }8,114
\displaystyle \text{R(16,16) = }10,796
\displaystyle \text{R(17,17) = }14,097
\displaystyle \text{R(18,18) = }18,106
\displaystyle \text{R(19,19) = }22,918
\displaystyle \text{R(20,20) = }28,634

which all fall nicely into current best-known intervals for these numbers. But these values are essentially nonsense. However, it is currently believed that \displaystyle \text{R(5,5) = }43 rather than 49. But there is no polynomial expression that yields 43 and the other lower already known numbers in the sequence of Ramsey Numbers evident from running long and large searches for such. Therefore, I believe that any meaningful approximate or asymptotic formula must be non-polynomial.

Therefore, my intuition says that there may well exist an exponential, non-polynomial expression for the Ramsey Numbers — perhaps similar to the P(n) function of Rademacher such as

\displaystyle p(n)=\frac{1}{\pi \sqrt{2}}\sum\limits_{k=1}^{\infty }{\sqrt{k}}{{A}_{k}}(n)\frac{d}{dn}\left( \frac{1}{\sqrt{n-\frac{1}{24}}}\sinh \left[ \frac{\pi }{k}\sqrt{\frac{2}{3}\left( n-\frac{1}{24} \right)} \right] \right)


\displaystyle {{A}_{k}}(n)=\sum\limits_{0\le m<k;\ (m,k)=1}{{{e}^{\pi i\left[ s(m,k)\ -\ \frac{1}{k}2nm \right]}}}.

But then I may be dreaming here because there need be no sequential connection between one of these numbers and the other — if each is a unique value in it’s own problem space.

What I find frustrating with papers on Ramsey Numbers that I have read is their lack of a more probing approach. We know that calculating Ramsey Numbers is NP-hard. One paper even suggested that this is Hyper-NP hard — but did not specify in what manner they meant this to be true. Most likely they were referring to the absurdly rapid exponential growth of the possible solution space. But where are the more basic insights into the nature of these numbers? Most of what we have now is not much beyond Paul Erdős work in the 1930’s!

In a recent paper on Ramsey Numbers, the physicist Kunjun Song, said: “Roughly speaking, Ramsey theory is the precise mathematical formulation of the statement: Complete disorder is impossible. or Every large enough structure will inevitably contain some regular substructures. The Ramsey number measures how large on earth does the structure need to be so that the speci ed (sic.) substructures are guaranteed to emerge.”

I think that we have yet to ask ourselves the deeper questions to get further along here. I am now researching the different ways in which the same questions may be asked, such as via Shannon limits of graphs and quantum algorithms to see what pure mathematical insight might be gleamed from these approaches. I am looking for good questions that, if properly phrased, should provide a road-map for further fruitful research.

© 2012 Kurt Lovelace – All Rights Reserved

This was the first week of my senior-level class with Professor Siemion Fajtlowicz, MATH 4315 – Graph Theory, and it was a blast!

The central question put to the class is when are two graphs isomorphic. There is nothing easy nor trivial about this question. It can be challenging to even distinguish that two simple graph representations are of different graphs, let alone of the same graph.

Also, we have been assigned the task of showing how one is to interpret the historically notable problem, first presented to and solved by Leonhard Euler in 1735, of the Seven Bridges of Königsberg in a graph theoretic manner. Euler resolved this question in the negative, but there is a lot more to it than that, as we will see in this article.

Along with this problem, we have also been assigned the Knight’s Tour problem which we are also to interpret in a graph theoretic fashion.


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