Erm, the magical plane and the ascended plane are not the same plane... That's what makes the AK so special, it draws power from both of them. We already know what causes the effect on the magical plane, at first its the incantation, then it can progress to silent casting.

I assume by different means of casting you do mean silently/no wand movement, cause I can't think of any other way (bar wandless magic, but I'll get to that later) that they could be cast. If so, they're already in use and I doubt anyone, especially in the Wizarding World, would try to fix something that isn't broken. If it works, don't fix it.

On the arithmancy note, I think that's how they would probably do it, though it'd be extremely complicated to get just the right effect because even the slightest mistake could bring out a totally different effect or could blow the spell up in your face. That's why people who invent spells have their own Chocolate Frog Card (going off the CoS game here - which is actually quite amusing) and are likely hailed as great inventors.

Wandless magic, I'd assume, is much less efficient than wands, else wands wouldn't be needed. Just think about it - a wand is just a focus for the magic to go through and be released, a hand is much bigger than a wand, and is a part of your body so it doesn't have a specific pool of magic to channel it through, cause magic is channelled through the whole body, not just the torso/head (my reasoning behind this is that auras are just as bright around the whole body, not just one part). That means that it can't be focused into a tight beam, making it more of an area effect spell spreading outwards than a spell as we know them.

Aekiel

 

We already know what Arithmancy "does", it's a form of divination using numbers and sigils to predict the future...

 

Arithmancy isn't a way to predict the future, this is the explanation I got:

Not quite predicting the future, though it is a form of divination, albeit a completely different one to Trelawney's class. I always thought of arithmancy as the foundations for going on to spell crafting, working out how a spell works then trying to create a new one based on that research.

Aekiel

 

Hmm... I suppose I was a little vague when i posted that, and half asleep too... here's what I get from my reference books (i've got a bunch of magical reference things on my bookshelf).

The text goes on with how to use this type of arithmancy, and i don't have the patience to type the rest of it out. But as we can see there is nothing here to suggest Arithmancy is a way to create spells. Even in the books there is no mention of Arithmancy being a component of spell creation - look at the spells Snape created as the Half-Blood Prince, there are no arithmantic equations - just the incantation.