The Justifiability of Irrational Numbers

I have been doing Algebra 1 this year, and I really enjoyed it - until I came to the chapter on irrational numbers. Now don’t be intimidated by my mentioning algebra/math. This is really going to be a very simple post. But I will warn you ahead of time, that you can’t get much out of it by skimming it. Get in, or stay out.

This post was inspired by a discussion I had with a clever young man, at a chess table at the 2009 Home school Book Fair. (Though I think I have had this conversation more than once, with more than one person.) Being multitaskers, we tried to have the discussion, with inserts from several other bystanders, and play chess at the same time. It was quite educational. I think it started with him trying to explain to me why one cannot know both the velocity of an object and the position of that object, and from there we got onto irrational numbers and the Theory of Relativity. (This post only covers irrational numbers.)

Irrational numbers are introduced with the Pythagorean theorem:

In other words: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

This is true for any right triangle. Example:

hypotenuse = c

1 leg = a

2 leg = b

So we translate that as follows: 4² + 3² = ?²

4 squared really means 4 times itself, which comes out to be 4 x 4 = 16

3 squared comes out to be 3 x 3 = 9

16 + 9 = 25

25 now equals the hypotenuse, squared. In order to undo the square, we must find the square root of 25. If you have any algebra, it is not hard to figure out that the square root of 25 = 5. (5 x 5 = 25) All very well and fine. But it doesn’t always work. What if you do this?

Should be pretty easy, right?

1² + 1² = ? ²

1² = 1 (1 x 1 =1) so 1 + 1 = 2²

? ² = 2²

Thus, the square root of two is the length of the hypotenuse.

How in the wide and wild world does one ever find the square root of 2?

It doesn’t seem so complicated at first, unless you have already been through it before, and are already thoroughly aware of the scandalous problem. But if not, it won’t take long to become so. There can only be so many numbers, right? 1x1 = 1 which is too low. 2x2 = 4 which is two high. It must be a fraction/decimal. But you could spend weeks trying to find what fraction works, and never succeed. In other words, the square root of 2 cannot be represented by a whole number or a fraction.

1.4142135623730950488016887242097 etc. is the closest you can get. But it is still too low. What is more, that number does not end with the 7, but goes on into eternity, in an endless string of numbers that NEVER repeats itself. There is no pattern. It is absolutely random.

1.4142135623730950488016887242097 x 1.4142135623730950488016887242097 = 1.9, with a repeating 9 that disappears into eternity. Not quite 2. The square root of two is called an ‘irrational number.’ What’s more, there are hundreds of other ‘irrational numbers,’ and unfortunate high-school students all over the world are expected to put up with the infamous incongruity and use irrational numbers in their calculations like any other number.

But in spite of the best algebra teachers available, the theory still does not make sense. It defies reason. How can a finite length, in this case the hypotenuse of the above triangle, be nonexistent? Because if it is not a whole number or a fraction, how can it exist? How can that simple black line have no definite length? It simply cannot be.

You can take a ruler and measure that mysterious black line. No, you won’t get an exact number, the ruler isn’t precise enough for that, but surely an exact number must exist. To say that it does not defies the imagination. How can that line have NO exact length?

I don’t know quite how it works, but I have an idea. Let’s back up several decades and think of it this way. If you have an object, for the sake of illustration let us say you have a rectangular eraser, than you can split it in half. You can split it in half again, and again and again. Theoretically, you can split it in half eternally. There is always that one piece of matter, and there is always one half of it to split. Once you get to the atomic level things might get more complicated, I don’t know. Leave it at that.

Now, what if you use a number? This might be slightly more difficult to fathom, because a number is not a piece of matter. It is more like an idea. Let’s use the number 12. Split is once and you have 6, split it again and you have 3, split it again and you have 1.5, split it again and you have 0.75, split it again and you have 0.375 and so on. We need not say that, theoretically, you can split a number infinitely, because numbers as we are dealing with them are already theoretical, and you can certainly split them infinitely. You can split the number 12 until your head splits, but I don’t aim to go that far. The point, that I have taken a vastly long time to make, is that a number can be split infinitely. Let’s move on.

This is a very simple number line that I have created with the help of Microsoft Paint, (a certain software that I have a love/hate relationship with.)

Now, stare at that beautiful piece of art and think about this very carefully for a moment.

IF YOU CAN SPLIT A NUMBER INFINITELY, THAN INFINITY EXISTS WITHIN THIS NUMBER LINE.

Did it click? If it didn’t than I’ll spend one short paragraph trying to explain it a little better.

There are an infinite number of points on that number line, because the points can always get smaller and smaller and smaller, (since you can cut them in half eternally.) So even though the number line has a finite length, infinity exists within it – on a very small scale.

If you don’t agree that there is infinity within a line, than you may as well stop reading right here. You can write me a comment, asking me to do a better job of explaining, or telling me I am dead wrong, and I’ll try to help.

If you do agree, than we’ll go on to the final statement of Part One.

Try to relate ‘infinity within a line’ with ‘an infinite number that is not a whole number or a fraction and never reaches the square root of 2.’ If you think very hard for about three minutes, you will probably be able to fathom how an infinite number can exist within a line. How the length of a line can actually be an infinite number. It all hangs on relating the two concepts I just mentioned. If infinity exists within a line, than it might just possibly be able to imagine how a number can be infinite within a line and never quite reach a whole number or a fraction that, when multiplied by itself, equal 2. In a way, the square root of 2 does not exist. Almost, but not quite. There is just no point on that number line that is the square root of 2. To me, at least, that begins to make a shadow of sense.

When I first came to this conclusion, with the help of the Physicist, my first objection, – I like making objections – which may possibly be what is going on in your mind right now, was that the square root of 2 goes on and on forever, but the line doesn’t. There may be infinity within the line, but the line itself isn’t infinite. I have no proof either way, but if you think about it, all of the numbers that come after 1.4 get very small very fast and it is only reasonable that they correspond with the points on the line accordingly. If that didn’t make any sense, and doesn’t help you, than just leave it alone and go on.

This is all conjecture – none of it is proof – but I have not yet even conjectured, much less proven, that irrational numbers are at all justified. All I have done so far is given an argument for the existence of an infinite number within a finite line. Which fact most mathematicians accepted several thousand years ago. It seems quite simple, actually, once you have thought of it, but it was very hard for me to fathom at first. Well, actually, I have done a little more. I have given an argument for the existence of an infinite number within a finite line never reaches anything that is exactly the square root of two. In other words, it never reaches anything that, when squared, that is, multiplied by itself, will equal 2.

By the way, before we go on, let me clear something up. If my repeated reference to a number that ‘never reaches,’ bothers you, let me put that straight. Most numbers don’t reach anywhere, but if you have an infinite number, it is rather difficult to speak of it without using the word ‘reaches.’ Since it is infinite, and goes on and on and on, it is actually, in a way, ‘going’ somewhere. Thus the word ‘reaches.’

But anyway, the conclusion we came to above is not necessarily an irrational number, I don’t think. An irrational number is not only infinite, it also, supposedly, never repeats itself.

Infinity within a line posed a problem to me for a while, but once you get over it, it becomes minuscule in the presence of this colossal problem. Because now that we have got over the argument that it cannot be, we are faced with the much more significant argument that it should not be. The math books try to tell us that that number that never ends and never repeats itself is random - absolutely and eternally random, and for me, that is far more difficult to fathom than infinity within a line. The universe is NOT random!

But this post is already four pages long, and even if there is anyone who has read this far, they will probably not be inclined to read any farther just now, so the rest of this will have to wait for a future post.

One more thing. If you read this post thinking, yes, of course, or I knew that all along, or just what I was going to say, and were sorely disappointed at being unable to participate in a controversy simply because there was nothing to argue about, than here are a few other points that might trigger a cyber debate.

Einstein’s General Theory of Relativity – is there an absolute by which to measure the motion of objects, or can we never discover whether the ball that fell from the train was traveling straight or curved or both or neither?

The Nature of the Universe – Is the Universe finite – or not? Is a finite Universe fathomable? What exactly do we mean when we say ‘universe?’

The Andromeda Galaxy – Can we read history in the light of a galaxy 2.5 million light years away? If that figure is accurate, than what are we really seeing? Can we look back in time?

I don’t necessarily have opinions on any of them, but I would love to have it out with someone and create an opinion on any or all of them.

THE SONG OF THE CHILDREN

This is a poem written by Chesterton. I think it is simply beautiful.

The World is ours till sunset,
Holly and fire and snow;
And the name of our dead brother
Who loved us long ago.

The grown folk mighty and cunning,
They write his name in gold;
But we can tell a little
Of the million tales he told.

He taught them laws and watchwords,
To preach and struggle and pray;
But he taught us deep in the hayfield
The games that the angels play.

Had he stayed here for ever,
Their world would be wise as ours--
And the king be cutting capers,
And the priest be picking flowers.

But the dark day came: they gathered:
On their faces we could see
They had taken and slain our brother,
And hanged him on a tree.

Amazing Grace

Take a minute to watch this awesome video. It really is amazing. But you have to watch it all the way through.

http://www.youtube.com/watch?v=YtrnB4F

Art

For the past several years I have studied dozens of artists and over two hundred paintings and have (finally) decided who my favorite artists are.

My three favorite artists of all time are Michaelangelo Buonarotti, John William Waterhouse, and William Bouguereau. I like all of them for the same reason - their paintings are realistic, and their subject matter is beautiful. Beauty is the number one quality that all art must have in order to deserve the title. The element of meaning that a picture contains is also important, but secondarily. There are plenty of 'meaningful' pictures today that are downright hideous.

Buonarotti, as my favorite Rennasaince artist, really represents my love for the Rennassiance painters, among them Bernini, Rembrandt, and Caravaggio. I like their colors, which are rather dark compared to more recent artwork - I generally prefer darker colors to bright ones - and I like their style. Art in those days was just breaking out of the medieval 'ice age' where all the figures are frozen onto the page and perspective is worse than a six year old's, but it retained the grace and elegance of the art that preceded it, something that more recent art has lost. The Sistine Chapel still has the half-frozen fairy tale look to it, but it is very much alive. And I think that, as a sculptor by trade, and thus being well acquainted with the human body, Buonarotti captured the art of making his figures look even more life-like than most other Rennaisannce painters.

Waterhouse represents my love for the Pre-Raphaelites. The Pre-Raphaelites are my favorite 'group' of painters. Their colors are a little bright, but I don't really mind. Waterhouse's subject matter is what is really appealing to me. He illustrated mythological stories, and loved to paint beautiful women from legends. Seeing that I can't hardly find a Pre-Raphaelite painting that I don't like, I have many, many favorites among the Pre-Raphaelites, and I recommend looking up some of their paintings, which are simply beautiful.

Bouguereau is maybe the most talented of the three when it comes to painting. Bouguereau's pictures are striking for their soft, smooth texture. They could almost pass for photographs. They are gorgeous.

Runners up are Jacques-Louis David, Frederick Leighton, Gian Lorenzo Bernini, Rembrandt, and Thomas Cole, among many others.

It took me a dreadful long time to decide what my favorite paintings were. The first one was fairly easy, the second one rather more difficult, and in all fairness I must say that I am still not completely sure about the third one. There are so many. Anyway, here they are, in order.

The Creation of Adam, by Michaelangelo Buonarotti. There is no doubt about this one. The power in this picture is awesome.

My second favorite is Napleon Bonaparte at St. Bernard's Pass, by Jacques-Louis DaVid. This is truly splendid. Not that I care that much for Napoleon himself, but the look on his face could change a world. This picture reminds me of Longfellow's poem, 'Excelsior.'

My third most favorite picture is probably Proserpine, by Dante Gabrielle Rossetti. This is a beautiful picture, but it really ties with several others for third place.

Five top runners up (in no particular order) are:

Tristam and Iseult, by William Waterhouse

David, by Gian Lorenzo Bernini

Hero Awaiting the Return of Leander, by Evelyn de Morgan

Innocence, by William Bouguereau

The Flagilation of Christ, by William Bouguereau

These paintings are what art is really supposed to be.

Contest Announcement

Hey. Hop on over to my sister's blog and see if you can win the contest she is having.

Cultural Literacy Contest and Goats

The Lady of Shallot

This is Loreena Mckennit singing Alfred Lord Tennyson's poem, The Lady of Shallot. I love this poem, and the way she sings it.

DAWN COLOR

This is a sestina that I wrote last week. No, I didn’t know what a sestina was either, but it sounded really neat when I looked it up, so I tried my hand at one. A sestina has six stanzas, with six lines each. Each line ends with one of six key words. It does not have to rhyme, but it can, if you think you’re that talented. I have an occasional slant rhyme in mine, which is purely accidental, but is kind of nice. Because I was also trying to write a sestina with a certain color theme - red - the words I chose were dawn, fire, sun, desire, blood and color. In the first stanza, the words are arranged however you want - after that it gets more complicated. This is the tricky part. The first line of the second verse must be the last line of the first verse. The second line of the second verse must be the first line of the first verse. The third line must be the fifth line of the first verse, and so on. It took me awhile to get the hang of this, so I made a table that might be helpful in case you ever feel like writing one. (It really is fun once you get started. If you ever do, I would love to read it!)

First Verse Second Verse Third verse
First Line dawn color sun
Second Line fire dawn color
Third Line sun blood desire
Fourth Line desire fire dawn
Fifth Line blood desire fire
Sixth Line color sun blood

The fourth verse will begin with blood. See the pattern. And so on…

That’s not all. The six verses are followed by a closing verse of three lines. All six key words must be included in this envoy.

Here’s mine.

I walked the edge of the newborn dawn
That wakened the world with fire
And I stood on the brink of the setting sun
In front of the world’s desire
And I realized that this is a world of blood
Because red is the only color.

And I realized that red is the only Color
In torches of silence at Dawn
In cardinal sunsets and roses of Blood
And wine-kissed lips in the Fire
In the sacrifice and the one Desire
When the eyes of the dragon burn like the Sun.

The morning is washed with the bleeding Sun
Because red is the only Color
In the burning gold of death’s Desire
And the blazing passion of Dawn
And the crimson jewels of the frozen Fire
Drenched in a poppies’ Blood.


The streets that ran red with the lover’s Blood
The sacrifice of the Sun
Because of the Tree of eternal Fire
Because red is the only Color
The ultimate Lie on the ultimate Dawn
And the apple of all Desire.

The ones who were caught in the one Desire
Were slain in the City of Blood
For the thing that brings the wind of Dawn
Is known as the Setting Sun
And they realized that red is the only Color
When they turned to face the Fire.

The sky sank into a lake of Fire
Burning in tears of Desire
And I realized that red is the only Color
And the world is made of Blood
And I turned my back on the setting Sun
And went out to look for Dawn.

But the Sun went free, and the Fire sang
And the Blood of the starlight was washed in the rain
And the Cursed Color of all Desire fled from the dying Dawn.

It doesn’t hang together very well, because of the color theme, so if you were thinking that you missed my message, don’t! I concentrated on imagery in this poem, not meaning. I personally like poetry to have a distinct, if subtle, message, but I was playing around with pictures this time, and I didn’t bother.

This isn’t actually my favorite kind of poetry. I don’t think it allows nearly enough freedom of expression. Even though it doesn’t rhyme, there are too many rules. It is probably as far as you can get from free-verse! I generally stick to something in between those two extremes. I like rhyme, rhythm, syllabic verse, and conforming stanzas, and I don’t tend to write without them, or to use white space creatively, but I don’t care for so many restrictions. Still it was fun, and I like exploring new kinds of poetry.

Namarie,
Raora

Utopia - The Land of Nowhere

In 1516, Sir Thomas More published his controversial work, Utopia, a book in which More's adventurer,Raphael Hythloday describes an imaginary land that he says he stumbled upon across the Atlantic somewhere in the New World which had just recently been discovered. Utopia is a perfect image of the ideal communistic world. Everyone lives in peace, everyone is equal, and there are a set of simple laws that hold the society together and cause everyone to work together as a community, thus making Utopia one of the richest and most successful countries in the world.

At first sight, Utopia is enticing. There are none of the evils that plague our own world. Everything works together perfectly. Crime is so rare that one may almost count Utopia to be free of it. The system works so well that no one need work more than six hours in the day. They pursue the arts and the sciences in their leisure and everyone has the chance to be educated. Everyone follows the law and go along with the way of things. They are rich, prosperous, learned, and - at least it seems so - happy.

As one goes deeper, there are things that turn one off. For instance, there is no such thing as individualism. At least, it is certainly discouraged. People are meant to be uniform products of the system. Educated and intelligent products perhaps, ‘good’ and ‘moral’ products perhaps, but no more. There is no such thing as private property - everything is shared among everyone. Everything that one produces must be handed over the ‘state’ to be shared out equally among everyone. One relies upon the system for everything. But then, one might say that, so long as the system works, it is well. One might say that through a kind of slavery the people have obtained true freedom. And then, one might say the opposite. That through the obtaining of freedom the people have become unwitting slaves to the system. An excellent system, maybe, but is it enough? Personally, I would throw my lot with the outside world with its grief and joy and risk utter slavery to obtain utter freedom.

To me, the Utopian’s alarming Epicurean philosophy is one of the worst aspects of their society. I have read little about their beliefs as yet, but their religion, or lack thereof, seems to me, to be nothing save a replica of that of the Romans. God(s) who are nothing save an image on the surface. When the cloak of religion is taken away, the starkness of their Epicureanism is truly frightful. Pleasure is their only goal. To follow after pleasure is their sole occupation. Thus their clever system that lessens their labor as much as possible. True, the Utopians follow only after pleasures that do not result in unpleasantness to anyone else. Also true, the Utopians follow only after ‘real’ pleasures. For example, there is no ‘real’ pleasure, according to them, in a piece of gold. One cannot eat it. They do not use money in their trade, so it has no way of giving pleasure. Hoarding treasure for its own sake is not pleasure. There is no wisdom in preferring purple cloth over plain cloth. If they are both equally soft, the one color does not give one pleasure over the other. One might say that there is good sense in the Utopian’s pursuing of ‘real’ pleasures. There is nothing else, with their façade of a religion. But one might say that is the ultimate reason why not. If there is nothing else, than pleasure will have to be enough. But what if it is not?

More seems to realize that such a revolutionary idea must remain nothing save an idea for a good time yet. He says that ‘this thing cannot come to pass until all men are good, and that, I think, will not come about for a good many years.’ The word Utopia means nowhere. But there were some who took the idea and left the last warning. The knowledge that men must be ‘good’ before Utopia can become a reality did not stop ambitious men such as Karl Marx and Vladimir Lenin. It is very probable that the whole terrible idea of Communism arose from More’s sixteenth century book, and there is little probability that it has left the world any better than it found it. There are plenty of controversial ideas in Utopia, of objectionable statements, of questionable conduct. But the question of whether or not Utopia would be what the world needs will never be answered. For that is the one ultimate flaw in Utopia. It is impossible to achieve.

I'm Back!

If there is anyone who still reads my blog - I'm back! During the holidays I took an extended break from blogging and I meant to start up again sooner, but I wasn't able to because of...
You guessed it! Goats! We had seven baby goats in one week! It was awesome!

The mother in this picture is Torfrida, named after Hereward the Wake's wife. She was the first to kid. She gave us triplets, but one of them died soon after it was born. The survivors were both girls, and we named them Luthien and Tinuviel, (from the Silmarillion). Luthien is the brown one and Tinuviel is the white one. (Luthien is my favorite!) We are bottle-feeding them twice a day.

Princess Leia, named after Star Wars, had the next babies. She originally had a boy and a girl, but the boy died a few days later. This is the girl. We named her Padme Amidala. I know, it is a little backwards - Amidala was Leia's mother in Star Wars....Oh well. She doesn't like us very much because she was sick the first couple days and we had to force feed her nasty-tasting medicine. I don't have that many pictures of her. Fortunately, she is smart enough to nurse from her mother, and her mother has plenty of milk, so we don't have to bottle-feed her.

Our Nubian goat, Brunhilde, (named after Wagner's Ring Opera) had twins as well, a girl and a boy. Both of them have survived; the girl's name is Freyda, and the boy's name is Odin. We are bottle-feeding them as well.

Aren't they cute?! We are having a great time with them! Hope you enjoyed the pictures!

Merry Christmas! (With A Capital C)

The Silmarillion and a Pre-redemption World

I received an interesting comment from a reader on my last post:

"I have just one little criticism; for Tolkien's about to be about redemption is a nice thought but I don't think thats what it is about. Tolkien's writings are more about the "deeper reality". He tries (and very effectively) to portray the battle of the spiritual world though fiction. The LOTR is a perfect example of how he illustrates the "deeper reality" through his "fictional" writing. But in fact his "fiction" is closer to reality than books that are about "reality." The idea that his writing is about redemption is thoughtful and hopeful, but not really logical."

Thanks for sharing your thoughts, Quin. I do believe that Tolkien meant to portray the battle of the spiritual world though fiction, and that he did it very well. That is one of the reasons that I love his books. But I also think that Tolkien meant his fictional world, in which, it must be noted, he purposefully created numerous parallels to the 'real state of things,' to be a pre-redemption world, as I think is made quite plain by the quote that I posted and by other similar allusions. And I will take the liberty of pointing out that I did back up my opinion with a logical quote, whereas you did not support your opinion with any evidence.

I will note that Tolkien made it very clear that he was not trying to write an allegory. To Milton Waldman he writes, "I dislike Allegory - the conscious and intentional allegory - yet any attempt to explain the purport of myth or fairytale must use allegorical language." Tolkien's works were not allegories, and therefore can not be expected to carry a specific meaning or symbolism for everything.

Nevertheless, Tolkien does not try to deny that certain aspects of his book represented and were meant to represent, if indirectly, certain Biblical truths. In his letter to Milton Waldman he makes it quite clear that the Ainulandile and the Valaquenta are a fictional picture of the Creation of the World and the Fall of the Evil One. The End of the First Age was the story of the fall of the Elves, who, are the central characters in the Silmarillion.

Tolkien goes on to say:
"In the cosmogony there is a Fall: a fall of Angels we should say. Though quite different in form, of course, to that of Christian myth. These tales are ‘new’, they are not directly derived from other myths and legends, but they must inevitably contain a large measure of ancient wide-spread motives or elements. After all, I believe that legends and myths are largely made of ‘truth’, and indeed present aspects of it that can only be received in this mode; and long ago certain truths and modes of this kind were discovered and must always reappear. There cannot be any ‘story’ without a fall - all stories are ultimately about the fall - at least not for human minds as we know them and have them."

Towards the End of the Second Age, we observe that the Fall of Numenor, or the Fall of Men, is in all probablity another allusion to the Fall in the Garden of Eden. There are similar allusions throughout his books, such as the quote I posted, and, while they can certainly not be pinned down as allegory, I think that their original sources can be tracked. What I am trying to say is that I think that it is sufficiently obvious to both of us that the Fall of the Elves and Men in the Silmarillion was a fictional allusion to the Fall of Adam, and thus, the entire race of 'real' men. It is equally obvious, to me at least, that Finrod's words to Andreth were a fictional allusion to the future coming of Christ. Assuming that that supposition is correct, it is only logical to conclude that Tolkien meant his fictional world to be pre-redemption. As I said in my comments on the quote from Morgoth's Ring, I posted that quote because I think it may be helpful in realizing that Tolkien had purposefully created a pre-redemption world, and because it was not a key theme in his books does not mean that he was trying to avoid it.

As an afterthought, I do encourage Tolkien fans to read his letter to Milton Waldman. It is very interesting and offers valuable insights into the motives behind Tolkien's works, and the nature of those works.

What Is Hope?

This quote, from Tolkien's mythological work, 'Morgoth's Ring,' has never been found in any other of his manuscripts. I think that this is one of my favorite quotes from him - especially the last paragraph.

In order to understand the following passage, I think it may be helpful to realize that Tolkien had purposefully created a pre-redemption world, but just because it was not a key theme in his books does not mean that he was trying to avoid the Redemption theme. Some people say that because Tolkien's books do not focus on the Redemption, like most of Lewis's do, they are not worth reading. I think that this passage might be the key to understanding some of the theology behind the pre-Lord of the Rings mythology, especially the Ainulindale. Also, as an afterthought, isn't it interesting to think that Tolkien was never able to finish the Silmarillion or most of his other First and Second Age books? I often wonder what they would have been if he had lived to complete them...

‘Have ye then no hope?’ said Finrod.

‘What is hope?’ she said. ‘An expectation of good, which though uncertain has some foundation in what is known? Then we have none.’

‘That is one thing that Men call “hope”,’ said Finrod. ‘Amdir we call it, “looking up”. But there is another which is founded deeper. Estel we call it, that is “trust”. It is not defeated by the ways of the world, for it does not come from experience, but from our nature and first being. If we are indeed the Eruhin, the Children of the One, then He will not suffer Himself to be deprived of His own, not by any Enemy, not even by ourselves. This is the last foundation of Estel, which we keep even when we contemplate the End: of all His designs the issue must be for His Children’s joy. Amdir you have not, you say. Does no Estel at all abide?’

‘Maybe,’ she said . . . ‘It is believed that healing may yet be found, or that there is some way of escape. But is this indeed Estel? Is it not Amdir rather; but without reason: mere flight in a dream from what waking they know: that there is no escape from darkness and death?’

‘Mere flight in a dream you say,’ answered Finrod. ‘In dream many desires are revealed; and desire may be the last flicker of Estel. But you do not mean dream, Andreth. You confound dream and waking with hope and belief, to make the one more doubtful and the other more sure . . .

‘What then was this hope, if you know?’ Finrod asked.

‘They say,’ answered Andreth: ‘they say that the One will himself enter into Arda, and heal Men and all the Marring from the beginning to the end. . . . How could Eru enter into the thing that He has made, and than which He is beyond measure greater? Can the singer enter into his tale or the designer into his picture?’

‘He is already in it, as well as outside,’ said Finrod . . . ‘For, as it seems to me, even if He in Himself were to enter in, He must still remain also as He is: the Author without. And yet, Andreth, to speak with humility, I cannot conceive how else this healing could be achieved. Since Eru will surely not suffer Melkor to turn the world to his own will and to triumph in the end. Yet there is no power conceivable greater than Melkor save Eru only. Therefore Eru, if He will not relinquish his work to Melkor, who must else proceed to mastery, then Eru must come in to conquer him. More: even if Melkor (or the Morgoth that he has become) could in any way be thrown down or thrust from Arda, still his Shadow would remain, and the evil that he has wrote and sown as a seed would wax and multiply. And if any remedy for this is to be found, ere all is ended, any new light to oppose the shadow, or any medicine for the wounds: then it must, I deem, come from without.’

-J. R. R. Tolkien, Morgoth's Ring

Latest Project

Alright, here is our latest project that I promised to post quite a long time ago. What do you think?

CAUTION: The following film is not intended for a pusillanimous audience.

The Death of Boromir

Strongjoy and I have been doing a little bit of filming with our digital camera lately, and this is our first production. We filmed it with some friends who came out for the weekend. We all had a great time acting out our favorite movie scenes in the woods, (I am not telling who I played. We made most of the costumes ourselves) then we fished in our pond and camped out on our land. We made s'mores, sang songs by the campfire, and heard the coyotes howl. Wish you were here.

Filming is a lot more fun than I thought it would be (and a lot more work!) I want to thank everyone who participated! We are nearly finished with another short film and I will probably post that when it is completed. Strongjoy and I actually do have a longer, more difficult one in mind for the future, but we haven't started filming yet. We are still working on script and costumes. I would tell you what it is, but everyone (including myself) who knows anything about it has been sworn to utter secrecy until it is produced, so I can't. The neat thing about this one is that Peter Jackson hasn't done it yet - in fact, we will be the first ones, so we are excited. Anyway, I hope you enjoy this one!

Talk About Imagination...

It is absolutely amazing what you can do if you have imagination. I really hope you check out this blog. These creative homeschooled guys read all kinds of great fiction authors, like Tolkien, C.S. Lewis, Arthur Ransome, Howard Pyle, Allen French and others, and have brought Middle-earth, Narnia and Ancient Britain back to life.

With only some very basic tools - fabric, leather, wood, clay, and of course, plenty of imagination, they have made and continue to make, all kinds of incredible costumes from uruk-hai outfits to real armor-strength chain maille to almost every kind of cloak you can think of - complete with helmets, swords, gloves, vambraces, sur-coats, standards, etc. They post numerous photographs, and provide detailed instructions on how to make the costumes. They have even created several short films, for which they sculpt remarkably life-like miniatures.

My brother and sisters and I used to do similar activities a few years ago and their blog has inspired us to start doing it all again! It is so much fun to act out all the great stories that you read about in all the great books. I have already finished making a few costumes, and we are working on some trial films. I may even post a picture of us in our Elven attire. We are having a blast! I am so grateful to Josiah and Jonathan for sharing their wonderful ideas.

Please check out their blog. I don't want anyone to miss out on this. Even if you think you are too old or too busy to do anything yourself, just look through all of the results of their creative imaginations and have fun! I am sure that it will be worth your time. I haven't enjoyed myself so much in quite awhile.