Setting the neck posture (line of neck or throat, not "base" of neck) parallel to the surface at address is "sort of" not really required for a) good beside-the-ball aiming, or b) straight stroking.
The requirement for accurately looking sideways along the ground to see where the putter face aims is that the golfer "aim the face" directly at the sweetspot of the putter head, and NOT that the golfer position the "eyeballs" vertically above the ball. So, aiming can be done accurately even if the eyeballs are not directly above the ball, but are inside the ball or even outside the ball. It is only IF the eyeballs are directly above the ball that the "back of the head" will also be "flat" to the surface and hence the neck will parallel the ground. Otherwise, not.
The notion from "robotic" putting instructors like Harold Swash (and nowadays his copier Paul Hurrion) is that UNLESS the "cervical spine" parallels the surface during the stroke, the golfer will have difficulty making a straight-back, straight-thru stroke. The underlying assumption is that a golfer NEEDS to move like a robot works for optimal stroke. This is a false notion that has apparent seductiveness only because people have very little idea about how their bodies actually are put together and work in movements.
ROBOTIC STROKES: The putting robot, whether Dave Pelz's "Perfy", or "Iron Archie" of the Putting Arc and VJ Trolio, or Harold Swash's "The Rack", all have ONLY ONE MOVING PART -- the shoulder frame plus arms plus hands plus putter assembly that is one structural unit that never changes shape. Although these metal contraptions are made to look somewhat like a human, they really are only one moving part on a base. The MOVEMENT pattern is determined ONLY by the way this assembly is fitted onto the base of the robot. The assembly has a hole in it between the two shoulders in the center of the shoulder frame and this hole is slipped onto a "pole" that represents the neck. Once the shoulder assembly is fitted onto this pole, the movement pattern is established by WHATEVER ANGLE THE POLE HAS TO THE SURFACE. The shoulder assembly can ONLY move (pendular swinging back and forth beneath the pole) in a plane that is perpendicular to the axis of the pole. Hence, if the pole is set parallel or horizontal to the surface, the shoulder assembly can and will move ONLY vertically to the surface. Whatever motion the shoulders have, the rest of the unit's structure will communicate that plane of motion throughout every piece of the assembly -- arms, hands, putter head -- and the putter head will also move in a vertical plane. That movement is a straight-back and straight-thru stroke. To make a SBST stroke, a knowledgeable teacher or coach would simply say: "Rock the shoulder frame in a plane that is vertical to the surface." (Pelz does not seem to know this and has never said it, so far as I am aware.)
Any OTHER ANGLE of the "neck" pole for the robot does not allow a SBST stroke, but generates a plane of stroke motion that is tilted with respect to the surface. ALL ROBOTS can ONLY make a stroke motion that is PLANAR, and it is not possible for any robot with only this one moving part (shoulder frame set on pole) to make an ARCING motion. For that to occur, the robot would require at least one other moving part, perhaps in the way the top fits to the lower base at the spine-pelvis connection, if there is one, so the upper "torso" of the robot as a whole pivots about. This is never done with robots.
Pelz's Perfy -- note the ball cap suggesting the robot is just a golfer human and that humans should move like Perfy. Also note that everything about Perfy from the base to the "neck pole" is fixed in shape and not a moving part -- only the shoulder assembly swings on the pole. Because Pelz erroneously teaches that the position of the hands in relation to the shoulders determines the stroke shape (the hand position is irrelevant and can be whatever the robot manipulator chooses in relation to the angle of the neck pole, as was explained to Pelz publicly at the World Scientific Congress on Golf in 2004), Perfy appears to have arms and hands but not a shoulder frame. In reality, the arms+hands assembly still has a hole in the top center and this hole is fitted onto a neck pole. When Pelz changes the hands inwards or outwards beneath whatever he calls the "shoulders" of this robot, he apparently has never noticed that the neck pole changes angle (rather unobservant and unscientific, really). Moving Perfy's "hands" simply alters the pole angle, as lifting the hands out away from the robot tilts the neck pole upward and generates a tilted-plane stroke that "looks like" a convex arc shape from the robot perspective, and setting the hands closer to the robot out of plumb with the "shoulders" simply tilts the neck pole downward and generates a tilted-plane stroke that "looks like" a concave arc shape from the robot perspective.
The Putting Arc's "Iron Archie" also has only the only moving part -- shoulder assembly on pole of neck -- and in the case of Iron Archie the angle of the pole of the neck is set ONLY to a 14 degree tilt up from horizontal, to give the illusion that the stroke makes an ARC when in fact it slides the putter head straight back and straight thru on a tilted plane and the face always stays square to the original direction of aim (albeit while rising and coming inside up the tilted plane in the backstroke and again in the follow-thru) and the face does not change orientation by opening or closing during the stroke.
Harold Swash's "The Rack". Note that the pole of the neck is set to horizontal -- any other angle generates a plane of motion that tilts to the surface. The neck pole is especially evident here. Swash at least understands robotic motion, whereas Pelz appears not to notice the one and only important setting that determines robotic stroke shape. No understanding of robotic OR human motion versus understanding robotic but not human motion -- this leaves the real quarry untouched: understanding both robotic and human motion as distinct.
HUMAN STROKES: A human is not a robot. Teaching a human to putt "like" a robot has superficial appeal but is fundamentally stupid. The orientation of the neck of the human does not especially influence whether the human moves the shoulder frame in a vertical plane, because the human body has many, many moving parts instead of just the one. It is quite possible and not especially difficult to move the shoulder frame in a vertical plane when standing erect at military address. This is a simple "lateral flexing" of the spine as when the human at military address reaches down the side of the left leg to scratch the outside of the left knee. As one bends the upper torso more and more forward at a putting address, it does not become a different or easier or more difficult movement, as the same muscles and joints are still working in about the same manner (the inner oblique muscles that attach the sides of the rib cage to the hips -- one side's inner oblique tugs the upper torso sideways towards the hip). So there is NOT an anatomical, biomechanical, or kinesthetic requirement that the neck be set horizontal to the surface at address. Does it help anyway? Not much if at all. It is probably best just to stand with relaxed balance more like Ben Crenshaw, as this allows better instinctive action since the golfer acts from a more habituated posture and balance without an unusual posture and balance as the base of the motion.
Ben Crenshaw at the Masters 1987, showing that an upright posture during the stroke (with line of neck tilted very much upward off horizontal to surface) has little to do with whether the shoulder frame action is vertical to the surface thru impact and down the line. A human is not a robot. A robot is not as good as a human. Among other things, a robot has no ability to read or aim the putt or to control distance or to make a mid-stroke correction or to choose the setup postures and ball position or even the stroke shape or pattern. A human who putts "like" a robot is not as good as a human who putts like a human. Pretending to move like a robot harms the sense of accuracy and touch in ways that putting from a more "natural" and "human" posture does not.
Incidentally, although Swash personally teaches that setting the cervical spine parallel or horizontal to the surface is an essential feature of making a straight stroke, he personally does not appear to use this setup posture for the stroke (he often says: "Do as I say, not as I do."), as shown in these photographs:
Harold Swash with his neck tilted up from horizontal.
Phil Kenyon, with neck set on a tilt upward to the surface.
Nor do students of these teachers set the neck parallel to the surface. Apparently, there is a difference between setting the "cervical spine" parallel to the surface and the neck itself above the cervical spine (whatever Swash means by that), as Padraig Harrington illustrates here in December 2006:
The cervical spine may be flat to the surface but the face and neck are hardly parallel to the surface.
Note: The aiming posture of the head and face and eyes and neck is required only for aiming, and after the aim of the putter face is checked, there is no real need to KEEP the same posture required for aiming as the one to use when making the stroke. The two postures can certainly be different, with the aiming posture a lot closer to setting the neck parallel to the ground (perhaps with the eyeballs vertically above the ball or close to this) and the stroking posture having the neck tilted up from parallel more (thus repositioning the eyeballs closer inward towards the feet and not directly above the ball but inside the ball). This is partly the explanation for why people like Scotty Cameron are not sound guides to head and eye posture -- they don't distinguish between aiming posture and stroking posture when they talk about the eyeball position, which relates ONLY to aiming posture, even though he is using as "data" the posture presented by pros during the stroke (and these pros aren't good aimers anyway). At any rate, Scotty Cameron does not even connect eyeball position with aiming -- he just states as a fact that pros in his studio have eyes 1.25 inches (on average) inside the ball, without any reason attached to the observation. He apparently has no clue what this relates to in putting or how it influences any part of putting.
Lee Janzen in Golf Magazine, Sept. 1998, page 37, supposedly illustrating correct "eyes over ball" setup at address but actually showing himself setting the eyeballs vertically above the ball with the neck horizontal and the face flat to the surface only in the first panel and THEN resetting his neck (apparently thoughtlessly without knowing he is doing this) on an upward tilt in the 2nd and 3rd panels and destroying his good aiming posture of the first panel. This is typical confusion.
YOUR QUESTION: Regardless of the above, if YOU want to set the neck parallel to the surface, here is how to do that;
Square up to the line of the putt so that the shoulder frame is parallel to the target line. Make sure the neck is perpendicular to the shoulder frame, and not tilted off to the right as is the case with "ballstriker neck" that afflicts so many top players due to excessive ball beating on the range in a "modified K" full-swing posture. Bend at the hips and upper back and neck simultaneously until a) the face as a whole aims straight down at the ball, and b) the eyeballs are directly above the ball. Both of these attributes must be present for this setting of the neck parallel to the surface to make any sense for putting. This posture is one of many that includes the straight-aiming of the face, so it will serve well for aiming beside the ball.
Other ways to set the neck / throat parallel to the surface:
The side pieces on glasses or sunglasses will aim straight vertically down when the eyeballs are vertically above the ball AND the face aims at the ball and putter head sweetspot.
The under-edge of the bill of the cap will aim at the ball when these two attributes coincide.
The plane of the face is "flat" to the ground, with the plane of the face being the plane that includes the two ear holes, the top of the head and the middle of the base of the neck, like two hands held beside the ears facing forward for better hearing.
The head biomechanically "nods" off the end of the spine as far as the neck muscles holding onto the back of the skull allow the head to nod -- as this posture is almost always equivalent to setting the plane of the face parallel to the surface (unless you are standing vertically in gravity on a serious slope).
Looking straight out of the face with the eyeballs and not looking somewhat down the cheeks ("like an old man peering thru the bottom lenses of his bifocals a la Benjamin Franklin"), the golfer's eyesight is aimed directly at the ball and sweetspot.
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