Errr...
It's also organised by properties and characteristics...
The left side of the table are metals. The right side are non-metals (to the right of that nice dividing line you refer to as the Silicon Staircase).
Check group 1... Lithium. Lithium is a soft metal, which you can cut with a knife. It reveals a shiny surface which tarnishes in air. Place it in water and it floats, then reacts violently, forming an alkali (Lithium hydroxide) and hydrogen gas, which combusts.
Next down the group is sodium. Sodium is a soft metal which you can cut with a knife. It reveals a shiny surface which tarnishes in air, even quicker than lithium's does. Place it in water and it floats, then reacts even more violently, forming an alkali (Sodium hydroxide) and hydrogen gas, which combusts.
Next down is potassium. Potassium is a very soft metal which you can cut with a knife. It reveals a shiny surface which tarnishes in air, almost instantly, in fact. Place it in water and it floats, reacts even more violently, melts due to the heat of the reaction, forming an alkali (Potassium hydroxide) and hydrogen gas, which combusts. It may, in sufficient quantities, explode in the course of this reaction.
Next down is caesium. Caesium is a very soft metal which you can cut with a knife. It reveals a shiny surface which tarnishes in air instantly. Place it in water and it floats, reacts even more violently, melts due to the heat of the reaction, forming an alkali (Caesium hydroxide) and hydrogen gas, which combusts. It will explode in the course of this reaction.
Last up is Francium. Francium does not exist in the natural world, as it has a half-life measurable in millionths of a second. However, in contact with water it will explode extremely violently (it has been postulated that 2.5g of Francium, dropped into the Pacific Ocean will explode with sufficient force to blow the Moon up).
All of the metals in group 1 have the same properties, increasing in reactivity as you move down the rows. This is why they were originally grouped together. Further, more recent physics experiments have shown that they all have a single electron in the outermost electron energy shell.
So in fact the Periodic Table IS organised by atomic number, mass number and elemental properties.
And hydrogen is still a metal - one electron in the outermost energy shell...