BeyerHardwick Method

The BeyerHardwick Method, abbreviated to BH, also known as 3Style, is an advanced blindsolving method based on the use of 3cycle algorithms. The fundamental idea behind the BH method is to use prememorized, move optimal, commutator 3cycles for all possible 3cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.
BH Corners
These are the different types of commutators you will "see" while learning BH.
Pure Commutators (8 moves)  are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.
A is always a 3 move insert B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move
To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.
There are only 3 A moves: R U R' R U' R' R U2 R' All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.
A9 (9 moves)  are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.
Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.
All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.
To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.
Orthogonals (10 moves)  are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.
To recognize them you must know 2 terms:
Opposites  when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR Note: We're only talking permutationwise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.
AnI or nI (Adj Noninterchangeable or Noninterchangeable  nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.
To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)
To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.
Cyclic Shifts (11 moves)  they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'
A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise
B is always a half turn and is not a opposite layer to either A or C Ex. A is R F so B must not be any form of B or L move
C is the reverse of A (not inverse) Ex. A is R F so C must be F R
Ex. [R F: U2] [F R: U2]
To recognize all 3 corners are on the same layer and are nI.
To do them figure out the middle piece of the cycle. Ex. UBL > LFU > BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.
If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.
Column Cases (11 moves)  they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.
To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.
Per Specials (12 moves)  a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.
Ex. [L U2 R' U2 L, U2]
To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL > DFL > DBR
BH Edges
These are the types of edge commutators you'll "see" while learning BH edges.
Half SlicePlanes (4 moves)  A special type of commutator that makes its interchanges on the same outer layer.
Formula: [A, B] where a A is the interchange and B is the setup to the interchange.
SlicePlanes (SP5, SP7, SP9) (5, 7, 9 moves  SlicePlanes are conjugated Half SlicePlanes commutators.
Pure Commutators (8 moves)  Same as BH Pure Comms.
A9/B9 (9 moves)  Same as BH A9s except B9s exists.
Orthogonals (10 moves)  Same as BH Orthogonals.
Learning Approach
Learning BH is a large task whether you'll opt for more speedoptimised algs some where down the line for some cases knowing the basic structure of a commutator [A, B] will help whether you'll look up a list or create the algs yourself since shortening an alg to [A, B] is alot easier to learn.
Creating the algs on your own is sometimes suggested as it will help you remember them more since you created them but looking up a list and not creating the algs yourself is fine but don't rely it on too much since that person's list is probably speedoptimised for them so if an alg is too hard to do try and make one for your self.
Note that BH or any form of commutators are intuitive meaning you don't need to generate the alg, YOU make them.
Whether you make them or not there's 2 routes to learning them: On the fly or learn the entire list before applying it on the cube.
Just like how F2L can be solved on the fly as long you know how to recognize certain cases or have a good sense for commutators you can actually not list every single case amd learn them rather you can do it while solving with no prior knowledge about a certain case just like you don't know every single F2L case but you can still do them.
Recognition
BH Corners
1. Pure Commutators
 2 corners in one layer and another corner in a different layer. (Ex. 2 corners at U and and the 3rd corner at D)
 The 2 corners in one layer are interchangeable
 There's a 3 move insert
2. A9
 Exactly like a Pure Commutator but with no 3 move insert (Not applicable to all A9s)
 Use elimination. (Ex. If it's not a Pure Comm with no 3 move insert then check if it's one of the other BH Corner Cases, if not then it's an A9)
3.Orthogonals
 All 3 corners are opposite and are all AnI to each other
4. Cyclic Shifts
 All 3 corners are in the same layer are are all AnI to each other
5. Columns
 Like a Pure Comm with no 3 move insert but the interchange corners are interchangeable with a half turn (Don't be confused with A9s)
6. Per Specials
 All 3 corners are opposite
 All 3 corners are mutually interchangeable
External links
 BH method for corners  Hardwick
 BH method for 3x3 edges  Hardwick
 BH method for 3x3 edges  Skarrie
 BH method for corners  Riffz
 BH method for corners  Jalocha
 edge cycles  lucarubik
 corner cycles  lucarubik
 Speedsolving.com: Thoughts about the BH method* [http://www.speedsolving.com/forum
 Speedsolving.com: Brian Yu's tutorial for BH corners
 Speedsolving.com: Chris Hardwick's explanation of BH 3x3 edges
 Speedsolving.com: Beginner BH ideas for big cube edges
 Speedsolving.com: Intermediate/Advanced BH ideas for big cube edges