Normally I'm not one for unorthodox builds, but this is one of those ones where it doesn't make any sense until you compare the results. A normal Lux build usually involves a Deathcap, some more random AP items, sorc shoes, etc.. This just doesnt work for me though. Lux's skill set is meant to carry, but her effectiveness ends when the teamfight does. I will explain more as we go on.

Please note: This is not a troll guide.
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make the enemy surrender to your awesomeness[/img]

I like to run almost all flat mana regen runes. Obviously magic pen reds, and a few flat cooldown reduction runes to help out with those ungodly long cooldowns. The reason for this is that a Lux's abilites are CRAZY expensive. Maxing mana regen for early game is the best idea to keep you going and keep you alive.

I use the standard 9/0/21. You want the cooldowns, mana regen, and the magic pen, It's as simple as that. Going 21/0/9 or going defensive spec is setting you up for disaster.

My picks are Teleport and Ghost. Being able to Teleport back to lane instead of having to run back to your lane makes everthing easier, or if you notice an opportunity to push, or if you see a champ with ripe for a laser killshot, then teleport is your best friend. Since you will naturally be so fast Ghost will make you impossible to catch unless you let yourself get CC'd.

Other good choices:
Claivoyance - A good spell, makes taking down runners easier. I wouldn't choose it over Tele but possibly ghost if you're confident
Flash - Hard to justify why not to take it in any case.
Clarity - In my opinion a necessity before summoner level 20.

You probably looked at the item build and thought "wheres the AP? Don't her abilities scale off of AP?" Yes, they do, but you don't even need it! Lets run some numbers:

A full AP build has somewhere in the neighbor hood of 700 or so AP
A more balanced approach has closer to 500 but a bit more survivability
My build maxes out at 180 with out a chance in the world of dying(If you can't get at least 15 stacks then find a different champ, If you are struggling in a game to maintain stacks and you have extra gold, trade in your mejai's for a deathcap)

Finales Funkeln with 700 AP at lvl 3 does 500 + .75(700) damage or 1025 damage before resistances. With 500 AP it does 875 before resistances. With 180 AP you deal 635 damage before resistance. Take about 100 away from that for most champions and you have 535 damage. For everyone but the tanks this is a quarter of their health, and lets be honest, when do you ever want to pump the laser into a tank? Never! You shouldn't need more AP than since both Bindings and Singularity will be hitting for around 300 each. Moving on.

Illumination deals 190 damage at level 18. Every damage dealing ability Illuminates the enemy but does not pop it, except for Finales which pops AND illuminates. So if you were to pop illumination in between every spell your damage would look like this
Bindings(300)>PING!(190)>Lucent(300)>Finales(535)>PING!(190). So you do a total of 1515 damage in one combo. For many champions, unless they are at full health this would completely decimate their health. If you're not alone (which you never shouldn't be), then your allies should have an easy time sweeping up the mess (which you don't care about because you can farm like a boss anyway).

Now comes the unorthdox; two phantom dancers and a manamune, there are several reasons why I do this:

Reason 1: Cooldowns and Mana Cost
As i mentioned earlier Lux's cooldowns and mana costs by no means short or low. In a build that maximizes AP you are pretty much useless other than a shield for almost 10 seconds. Now imagine having an ability that you could use to deal damage effectively when you are on CD. Oh, autoattacks, duh. If you correctly max out your tear and manamune then you should end up with approximately 130 attack damage. You also gain 60% crit chance from the phantom dancers meaning that more often than not you are hitting for 260 twice a second for the 10 seconds while you are CD.

Reason 2: Illumination
With Lux's normal attack speed it is nearly impossible to pop every illumination proc. However with the increased movement and attack speed you are able to hit almost, if not all of lit up enemies. When your build reaches its end it is possible to pop 10 illuminations before they fade. If it is creeps then you just earned 200 gold in 5 seconds without using hardly any mana.

Reason 3: Pushing
Lux without any attack speed has a hard time killing even the outer turrets by herself. Lux with a little bit of extra attack damage and a lot of extra attack speed melts a turret like Ashe. Coupling this with Lucent and Finales, Lux becomes a nasty split-pusher. If you have ever played Lux and tried to push a tower by yourself with a straight AP build, you understand that its like a mosquito attacking the Empire State building. I mean, eventually you will knock it down, but the bot lane would probably be able to make it all the way up to top before you got it a quarter of the way down.

Reason 4: Escape
If you miss the snare you are screwed. Unless your move speed is 550 AND you have ghost. The only person that can ever catch you is Yi, and thats only if he pops his ult, ghost, and buys a ghostblade and pops that as well. So not only do you have the ability to hit people from far away with your abilties, but also up with your autoattack and still be able to get away. Its like running away from a bear with your friend, just make sure you are faster than they are. Except instead of letting your friend get eaten by the bear you can turn around and shield your friend from the bear, slow the bear down, snare the bear, and shoot the bear and any other bears following with a laser. Catch my drift?

Reason 5: Positioning
Light Bindings is great, long range and it can snare twice, but it can be a ***** and a half to land with minions around. The crazy movement speed makes it a breeze though. They are fighting at your turret but all their minions are in the way to snare them? Run through the jungle and come up behind them. You will get there before they even know you left.

Note: Some who have tried this build prefer to have a Deathcap instead of Bashees (with less impressive results). I personally don't think glass cannons are the best idea, but if you think you can pull it off, I admire your balls.

Lucent Singularity first, always. This is what you use to farm, and is more effective for early game battles than Bindings are. In fact, I usually dont even get Bindings until level 4 most of the time. Lucent Singularity is your bread and butter spell for the first 8 levels or so. Levels 1-6 go E>W>E>Q>E>R from there you want look at whats happening, If your team is dying level up the shield first, If you need more damage level up Lucent first (I almost never have situations where i will level up W before E). Bindings should stay at level 1 until you have both your shield and Lucent at level 5.

Pros:
High Mobility
Abilty to Split-push effectively
More effective usage of passive late game
Higher Chance of escape
Easy repositioning
More damage over time

Cons:
Lower burst damage potential
Lower shield strength
Needs to be closer to the fight

Lux's skill set in my opinion is ridiculously overpowered. Add in a good auto attacking ability (Not like a real carry but still good nonetheless), and in effect this build turns Lux from a support/mage to a hybrid carry, and actually increasing her already amazing utility. This also makes her harder to counter build against, similar to other champions like Jax and Nidalee.