On Saturday, Richard Apparated to his lesson with his mentor in an excellent mood. He assumed it would be another training session devoted to Levitation Charms.
Madam Marchbanks was, as usual, waiting for the young man in her armchair by the fireplace. On the small table lay the wand Richard used for practice.
"Good morning, Tutor."
Richard first reached for the training wand. He was just about to begin casting the Levitation Charm when the old witch emerged from her reverie.
"Good morning, Richard. I have noticed that you find our lessons dull."
"Not exactly dull, but rather too monotonous."
"Well then, I suggest we add some variety. Watch closely, my boy."
On the table before Madam Marchbanks stood a large crystal dish filled with fruit. With a flick of her wand, she levitated one apple from the dish onto the table.
The old witch pointed her wand at the apple, gave it a small twist, and pronounced clearly:
"Geminio."
Before Richard's eyes, a miracle occurred—the apples became two. Outwardly, they were indistinguishable from one another.
"Is it an illusion?" the boy asked distrustfully.
"No, Richard, it is a copy. With the Geminio charm, one may create a duplicate of any object, food, or drink. However, copies of magical objects will prove unstable and will not retain magical properties."
"That can't be!"
"Here is the apple," Madam Marchbanks gestured toward the table. "You may eat it. It is perfectly safe. It will never disappear—unless it rots."
"No!" Richard stepped back. "That's impossible!" he said with profound shock. "How can that be?!"
"Heh-heh-heh!" A rasping laugh burst from the elderly witch's chest. "Oh, Richard, if only you could see your own eyes. I have rarely witnessed such astonishment. Of course, this is not the simplest charm—it is studied in the sixth year as an elective in Charms—but it is not as difficult as one might think. And the benefit it brings a wizard is enormous."
"I don't believe it…" Richard muttered quietly. "Am I truly witnessing the Banach–Tarski paradox with my own eyes? Thousands of students around the world have driven themselves mad trying to understand how, in theory, something could be created from nothing—and here, a simple flick of a wand produces a duplicate. An apple out of nothing!"
"Heh-heh! Richard, I have never seen such a vigorous reaction to a Doubling Charm. What paradox did you mention?"
"Ma'am…" Astonished, Richard could not tear his gaze from the apples. "Mathematicians among ordinary people have theoretically proven that it is possible to obtain a copy of an object out of nothing. If you take a sphere and divide it into five parts made of non overlapping points, the sphere can be doubled. Any two bounded subsets of three dimensional Euclidean space with non empty interiors are equidecomposable. One plus one equals one. Infinity plus one equals infinity. Infinity minus one equals infinity. Between zero and one lies an infinite number of numbers…"
"Oh, Richard, I see you have not been wasting your time and have taken an interest in numerology!" Madam Marchbanks exclaimed enthusiastically. "Most commendable. And those Muggle mathematicians are quite the tricksters, if they managed to prove the existence of doubling. Amusingly enough, you guessed correctly. To cast this spell, one must concentrate on the object and select a single point upon it. Then you rotate your wand to the left and mentally mark five additional points anywhere on the object. After that, you mentally divide the object into five parts and then reassemble them into two complete items."
"I am calm… I am calm… I am perfectly calm…" Richard whispered under his breath. "So what if wizards casually use the Hausdorff–Banach–Tarski paradox? Magic can do anything… I really should not just memorize that fact, but accept it…"
"So then, Richard, would you like to learn this spell?"
"Yes, of course, ma'am!"
Richard reminded himself that he was a Lord and must always maintain composure. Though he could be forgiven—after all, as a transmigrator he had been a lord for just over a year.
"But, Tutor, if this spell allows objects to be copied, then how have wizards not collapsed their own economy? After all, the most obvious idea would be to duplicate money and spend it. That would inevitably lead first to rampant inflation and then to the collapse of the entire economy, forcing wizards to revert to barter—if even that."
"Richard, first of all, all wizarding coins are enchanted," Madam Marchbanks explained. "After being minted, they are soaked in an inexpensive potion that turns them into weak magical objects. Thus, any copy of a Knut will inevitably be destroyed. There is a simple way to detect such counterfeits—the counter-spell Finite. Using counterfeit money carries punishment."
"Mm. I see."
"And another thing, young man. In your first year at Hogwarts, in Transfiguration, you will be taught the five Principal Exceptions to Gamp's Law of Elemental Transfiguration. One of them states that food cannot be created from nothing—but it may be multiplied or summoned with Accio. Water and certain other liquids, however, may be created."
"That aligns with the mathematical models of the paradox," Richard agreed.
"There are additional nuances, but you will learn about them in school," the old witch continued.
"Not now?"
"The second exception—magical objects cannot be transfigured. The third—you cannot transform an immaterial entity such as a thought or a ghost. The fourth—Transfiguration cannot create life. The fifth—you cannot create money, precious stones, or precious metals."
(End of Chapter)
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