I own nothing, on this story, other them my concepts.
Until he was five years old, Harry didn't know his own name.
It was always "Freak", or "Boy".
At five with primary school beginning, the use of his name was needed, or his relatives would have been questioned by Social Service, so a month before school started, Petunia Dursley began calling him "Harry".
School was a blessing and a curse for young Harry, his cousin Dudley, used his size to scare any child that got close to Harry, he even begun a game of "harry hunting" with his friends.
With this kind of incentive, Harry looked and found a place where his cousin wouldn't go after him, the school library during recess and Surrey Public Library.
With his quiet and polite ways, Harry found himself safe inside the libraries, after all Dudley stayed away from them.
In the libraries Harry found himself attracted to some subjects, like basic geometry and math, to a point that at his seventh birthday, he was looking at the more advanced calculus.
Even so, during class Harry tried to show as little as he could to not be punished for having better grades them his cousin.
Being an observant child, he noted that Dudley was only reprimanded when he used the word "magic", close to his parents, like it was a bad word.
As Vernon would say: " Don't say this freakish word, only crazy people use it!"
This and the fact that strange things occurred around him, made Harry sure that he was magical.
After his revelation, he used August and September, to have a semblance of control of his powers, at first he needed strong emotions to do anything, but with time he discovered that with a focused will and a calm mind many things could be achieved.
The first thing he did to discover how to use his abilities, was to try and shape one of the elements, with his best results coming from shaping sand, at a river bank close to the public library, into simple forms, within a week he could use the sand as a form of meditation, directing it movements under his will and feeling the grains move into patterns without needing to look at it.
With his curios mind Harry took a cup of sand hidden in his to big trouser pocket, to his cupboard to be able to do his experiments at night, while his relative slept at Friday and Saturday, that way none would be able to notice his late nights.
His first discovery was that magic easily followed some geometrical shapes draw with sand on his cupboard floor.
With the triangle being the most simple shape he used it to write numbers, and expressions.
1) To define a triangular basis, he used Sierpinski triangle, each level defining a power of three times the number of triangles external to the central triangle.
2) The negative numbers are written with the base triangle having their inner filled triangles swapped by border only triangles and the opposite for the border only triangles.
3) To write the inverses of the bases the multiplication triangles must be pointing to edges and not to the vertices.
4) Sums are done by aligning the empty point of a base triangle to the edge opposite to the empty point of the next base triangle.
5) To write a multiplication he would draw a filed triangle and point the summation lines to two extreme edge's points of the filled triangle.
From his studies, Harry knew that most of the sand was formed of quartz dust (silicon dioxide, or Silica), to remove the impurities, he toke some clay from the art's class at school, willing that the small amount, he toke, would be forgotten.
He shaped the clay into three bows at the bottom of one he draw the expression:
[(28) + 2x(16)]
On other he drew the negative form of the expression, to finalize the bows he forced the clay to expel humidity to dry the clay into ceramic.
During the next week Harry brought more sand from the river bank, enough to fill one of the bows and some more to draw the number [-3, inside the border only triangles, he put the three ceramic bows, with the one without an expression drawing, full of sand.
Harry closed his eyes and begun to will his magic to the filled triangle at the middle, slowly filling the sand drawing with a deep dark green glow, so soft, that anyone looking at the cupboard wouldn't be able to see a difference.
When he felt that he couldn't fill the drawing more, he begun to will the sand, in the full bow, to separate itself into, quartz dust and impurities.
The moment that he willed the purification of the sand Harry felt something new, the drawn sand begun to pull energy from the ground that wasn't his.
He opened his eyes, looked at the bows. They were glowing a soft green and the sand was flowing slowly from the full bow to the the other two, the bow with the positive expression was filling with a white powder, while the other was a coarse gray powder.
When the sand's bow was empty, Harry willed the silica unable to leave the bow and put it away in a dark corner of the cupboard.
After vanishing the content of the other bow to the house foundation, he brought his mattress back to sleep.
During the next week Harry begun to compare, how sand and silica reacted when exposed to magic.
While sand could channel magic well he felt the silica pulling not only his magic but the ground magic as well, forming small silica clusters in it. Pulling the magic back to use, unmade the clusters.
To get more clay, Harry went again to the river bank and used his magic to extract enough clay to make more recipients. Making sure to extract only hydrated alumina and silica mixture for his clay.
In his cupboard, Harry made sure to make bows for Alumina[2x(27) + 3x(16)], Titania[(48) + 2x(16)], Silica[(28) + 2x(16)], Hematite[2x(56) + 3(16)] , a vase for water[2x(1) +(16)] and the respective negatives for each, that way he could purify his working components and study their interaction with magic.
During his studies, Harry noted that the water had a different form of energy, similar but still different from his and the soil energy, needing a lot more energy to affect, making the work with water be more efficient by manipulating the other elements e making the fluids be expelled, without manipulating them.
The same for gases that need even more energy, because they have more freedom them the fluids.
Other them that Harry discovered that:
Silica accumulate any magical energy that is fed to it in clusters;
Titania under exposure to magic align itself in long fibers and them push the feed magic from itself on the other side of the filament;
Alumina has the opposite effect, instead of pushing it pulls magic fed to it, accumulating inside of the filament, and dispersing the magic when not charged;
Hematite neutralize any magical energy that is fed to it in the middle of the formed clusters;
Mixture:
(Siti)Silica-Titania form small clusters that emit controlled amounts of magic when in contact with a charged medium;
(Sial)Silica-Alumina form small clusters that absorb controlled amounts of magic when in contact with a unsaturated medium;
(Heti)Hematite-Titania form a film that impedes magic to be pushed through it;
(Heal)Hematite-Alumina form a film that impedes magic to be pulled through it;
(Tial)Titania-Alumina form a filament, that will emit magic from the end opposite to the end where magic is absorbed;
(Hesi)Hematite-Silica form clusters that filtrate injecting magic intent and overrides it with the stored intent.
Interesting enough after the two weeks of studding the components and their mixtures, Harry could clearly see that, with an adequate design he could build artifacts able to have spells, but first he would need to create the proper rite to turn the powders into crystals and glasses.