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Threshold (1997)
Sara Douglass

This is another fantasy book in which mathematics is seen as a sort of magic, but in this one it is specifically a particularly evil, cold and inhuman form of magic, in contrast to other less formulaic sorts of magic. There are the Magi, who are mathematicians, interested only in communing with "the One" through the Infinity Chamber in the building called Threshold. On the other hand, there are the Elementals who are more interested in things like life, love and beauty.

Of course, I worry that this sort of thing will perpetuate the stereotype of mathematics as an inhuman endeavor, of interest only to people who cannot stand the beauty and uncertainty of real life (a viewpoint with which I strongly disagree). Moreover, I'm not sure what to make of the gender issues addressed by the book. Certainly, in the book it seems that Magi are exclusively male, but I am not certain whether this suggests (a) women cannot do math (b) women are not interested in math because they are too full of life or (c) women are only excluded by sexism. In any case, the love between Elemental Tirzah and her Magus slave owner Boaz which is the main focus of the plot may serve to lessen any harm done by these suggestions.

(quoted from Threshold)

"Anyway, over generations there grew among the nobility a taste for mathematics, and eventually this taste solidified into a caste. Men only, for they claimed that women did not have the agility of mind to embrace the myriad complexities of numbers and forms. As I told you on your first night, Tirzah, the Magi, as these mathematicians came to be known...command power through contemplation of the One, and of all numbers and forms that the One generates.

"Gradually, their power and influence increased. The Magi loathed the magic of the Elementals, because they said it was unpredictable, reliant on chance and the whims of the Soulenai. Their magic, they claimed...was powerful because of its verdy predictableness and because, once its rules and parameters were understood, it could be manipulated to the Magi's needs. They work their magic according to set rules! Tables! Parameters! Can you imagine that?"

(quoted from Threshold)

I cleared my throat and read, praying my voice would remain steady and not irritate him with a stumble.

One, three, nine, eighty-one. A form in itself. Three lines of three, nine lines of nine, the square of beauty, breed into more beauty. Life is numbered from conception to death, rising from and declining into the One. There is beauty in numbers. This beauty is called Regularity, and its essense is Predictability. Everything else is unworthy. Eighty-one, nine, three, One. Life is numbered, all elements of life can be reduced to numbers, life is nothing but the predictability of numbers. There is nothing but numbers. Nothing. Nothing but the One.

I stopped. I could go no further. Tears filled my eyes.

"I was nine when I wrote that. The age of beauty I think, for nine is a special number in itself. ....Tell me what you think of it, Tirzah."

I answered trughtully. "I find it sad, Excellency."

"How so?"


"I find it sad, Excellency," I said slowly, "that a boy so young should find life so sterile."

This viewpoint of Tirzah's, that it is sad when someone is interested in mathematics because math is "sterile" is one that worries me. I must admit, I do find beauty in mathematics, but it goes beyond predictability (which does sound somewhat sterile) into intricacy, complexity, depth, symmetry and other words which are often used to describe the beauty of works of art as well. Moreover, I do not think that my appreciation of mathematics comes at the expense of my ability to appreciate other things in life.

BTW What did the author intend us to get out of the sequence (1, 3, 9, 81)? It looks like powers of three, except that the third power is missing, or is it the squares of powers of three with an extra term of 3 thrown into it? Either way, it doesn't quite make sense to me.

(quoted from Threshold)

"One day I heard two Magi briefly mention the numbers one, three, five, seven, eleven. They are another progression, perhaps." I prayed Boaz would not read the lie in my voice....

"You have given me only five numbers, and in themselves they do not make a progression. What follows the eleven?"

"I do not know, Excellency."


"There is only one thing those numbers have in common."

"Yes, Excellency?"

"They are all incomposite numbers, except the One, of course, which exists outside and beyond the others."

I did not have to pretend confusion. "Excellency?"

He sat forward, so I could see his face more clearly. "Incomposite numbers are those which cannot be factored, they cannot be divided except by themselves or by the One. They are thus indivisible."

"Then they would hold a special relationship to the One, Excellency."

"You are very good, Tirzah," he said softly, his eyes keen, and I thought I had gone too far...."And you are correct. Incomposite numbers hold a very special relationship to the One. They not only have a direct relationship with the One, they are in a sense different expressions of the One."

"So, from eleven the next would be thirteen? Then...seventeen?"

"Precisely, Tirzah. And then on infinitely. Perhaps we will make a Magus of you yet."

Although the book generally continues with the dichotomy that math is bad, lifeless magic, towards the end we find that there is even some mathematics in the Elemental magic.

(quoted from Threshold)

"Threshold is a mathematical formula expressed in physical form to provide a path into Infinity. The Song of Frogs is a mathematical formula expressed in music to provide a path into the Place Beyond."

"That's wonderful!" I cried. "Do you mean that if we learn the formula then we can travel to and fro between this world and the Place Beyond?"


"No...What I said" - and his voice was tight with strain - "was that the Song of the Frogs is a mathematical formula that will provide a pathway to the Place Beyond....but there is a catch, as there was always a catch with the Threshold formula -- although none of the Magi minded about that."

"And the catch is..." Isphet asked quietly.

"It's a one-way trip," Boaz said.


"Can you teach us this Song?" Yaqob asked. "Can you show us how it works?"

"I can't see why not. Listen..." and Boaz expounded the mathematical properties of the Song of the Frogs in exquisite detail.

None of us understood a word he said.

"Boaz, can you explain it in simple terms? None of us have had any training in the mathematics. And what you say..."

Boaz frowned. "That was the simple explanation. But I will try to put it a different way."

All we learned was that only someone with a mathematical background could understand this cursed Song.

Well, I can certainly identify with Boz when he sayd "that was the simple explanation." I've felt that way explaining math to students and others. And, I'm glad that the conclusion is that only someone with a mathematical background can understand it, because too many people believe that it requires some sort of innate ability. But, yes, some mathematics is difficult to explain to someone who does not have a great deal of prior experience. (Is that really unique to mathematics? I would think that most subjects get to such a point.)

Anyway, this is a nice fantasy novel with a definite current of mathematics running through it. If I was not maintaining this website and therefore viewing myself as having the obligation of evaluating and criticizing the role of math in each book, I think I would just have enjoyed it. But, in as much as it says something about the author's opinion of mathematics, I worry that she may not recognize how beautiful and full of life mathematics really is. It is not, as the book suggests, the antithesis of such humanistic values.

More information about this work can be found at
(Note: This is just one work of mathematical fiction from the list. To see the entire list or to see more works of mathematical fiction, return to the Homepage.)

Works Similar to Threshold
According to my `secret formula', the following works of mathematical fiction are similar to this one:
  1. Mathemagics by Margaret Ball
  2. When Women Were Dragons by Kelly Barnhill
  3. Your Magic or Mine by Ann Macela
  4. Counting the Shapes by Yoon Ha Lee
  5. Mathemagics by Patricia Duffy Novak
  6. Black Numbers by Dean Frank Lappi
  7. A Logical Magician by Robert Weinberg
  8. The Spacetime Pool by Catherine Asaro
  9. Coyote Moon by John A. Miller
  10. The World We Make by N. K. Jemisin
Ratings for Threshold:
RatingsHave you seen/read this work of mathematical fiction? Then click here to enter your own votes on its mathematical content and literary quality or send me comments to post on this Webpage.
Mathematical Content:
3/5 (1 votes)
Literary Quality:
3/5 (1 votes)

MotifEvil mathematicians, Female Mathematicians, Music, Math as Cold/Dry/Useless, Romance, Religion,

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Exciting News: The 1,600th entry was recently added to this database of mathematical fiction! Also, for those of you interested in non-fictional math books let me (shamelessly) plug the recent release of the second edition of my soliton theory textbook.

(Maintained by Alex Kasman, College of Charleston)