Eruvin 5 The unfenced courtyard and a mathematics teaser

We have learnt that although מדאורייתא (biblically,) an area enclosed on three sides is generally considered a רשות היחיד (private domain) as far as the laws of carrying on shabbos are concerned, there is a rabbinical requirement to mark or enclose the fourth side in some way.
It is important to note that the biblical rule could have both stringencies and leniencies associated, a subject I hope to discuss in a later post.
The leniency is that at least on a biblical level, one is permitted to carry within this area, or from this area to an adjacent private domain, without restriction.  The stringency is that if one carries from this area to a public domain, one would be liable for biblical level shabbos desecration, with all its ramifications.
The rabbinic requirement to enclose or mark the fourth side limits one’s ability to carry within that area or from that area to the adjacent רשות היחיד  without doing so, but probably does not affect the biblical prohibition against carrying from it to the רשות  הרבים.   
Until now, we have focused on a מבוי, or narrow street, which requires only a לחי (pole) or קורה (beam) to mark the fourth side.
What happens with an unfenced private front-yard or garden, either belonging to the owners of one house, or shared by various houses?
Does this also need to be enclosed, and if so, is the solution that works for a מבוי also sufficient for such an area?
On the one hand, this area is less public than a מבוי and more similar to a private domain by its nature, so perhaps Chazal didn’t see the same need to make it more distinguishable from the public domain.
On the other hand, it still shares an open fourth side to the public domain, or at least to a כרמלית ( open area not busy enough to be a public domain, but treated by Chazal with the stringencies of both public and private domains.)
On this daf, we see that there are strict rules defining the מבוי  that may be permitted by just a לחי  or קורה . Otherwise, it is considered a חצר (courtyard) and is actually treated more stringently!
1.       Its width needs to be narrower than its length, the width being the dimension only enclosed on one side, as opposed to the length which is the dimension enclosed on both sides.
2.       It needs to have houses and courtyards open to it. The Gemara (Shabbos 130b and Rashi) understands the later to mean at least two courtyards that each have two houses open to them.
As such, it seems clear that both a shared courtyard and a private one certainly do not meet the later criteria, and might sometimes not meet the former one either.
It seems to follow from here that at least the shared courtyard would definitely be treated stricter than the מבוי, and with the argument in favor of leniency for a less public area to be treated more leniently disregarded, in the absence of precedent to  the contrary , it seems that this would also be the case with a private front-yard or garden.
What precisely is required in order to be able to carry in such an area will hopefully be the topic of a later post as the sugyos develops.

There is a מחלוקת (dispute) on this daf between Rav Yosef and his student, Abaya regarding the minimum length of a מבוי.
Rav Yoseif holds that 4 טפחים (handbreadths) are sufficient, whereas Abaya requires 4 אמות (arm-lengths.)
Abaya attempts to prove his point from the above rule that we learnt- in order to be considered aמבוי  as far as the more lenient requirement for a לחי  or קורה, there have to be at least 2 courtyards that open to it.
As the minimum width of a פתח  (opening) is 4 טפחים  (the maximum being 10 אמות,) it is impossible for a courtyard to share one with a מבוי  that itself is only 4 טפחים long, without the entire length being open and thus disqualified .
The opening can also not be along the width that is already closed, as the width may not be wider than the length!
Rav Yoseif counters that one opening could still be possible on each side, if it is in the corner between the length and the width.
Rashi explains that this could be made of a 3 טפחים  gap along the length PLUS a 1 טפח  opening along the adjacent wall of the width, making the minimum 4 טפחים in total.
Tosfos , as well as other Rishonim make the rather strong observation that Rashi is not being precise, as the true entrance would then be marked by the diagonal between the enclosed part of the length and the enclosed part of the width, which mathematically (by pythagorus) will be the root of 10, still below the minimum width of 4 טפחים  !
Is Tosfos accusing Rashi of being unaware of basic mathematics such as the theorem of Pythagoras? Absolutely impossible, as there are various sugyos which mention this, approximating the root of 2 with 7 over 5 (See sukkah 8a for example)
It is also very simple for any mathematical layman to measure such a diagonal and see that the diagonal is much closer to 3 than 4.
As such, it seems clear that Tosfos understood that Rashi was aware of this discrepancy but deliberately chose to ignore it and be happy with an approximate minimum with  slightly more than 3 in place of 4, something that seems rather odd.
We have seen elsewhere that the Tosfos have pointed out that Chazal themselves were not always precise with their measurements (see Eruvin 13b for example) , but this was a question of rounding to the nearest integer, not rounding down more than a  half  and resulting in a major leniency.
It thus seems more likely that Rashi did not measure the entrance from the diagonal, but from the imaginary wall that would exist in the corner if the 3 plus 1 handbreadths were closed.
This would be a rather substantial מחלוקת  with a huge נפקא מינה (practical ramification) regarding the status of the area in-between this imaginary boundary and the diagonal as well as whether a bent opening like this is valid.
It is also clearly not the way Tosfos understood Rashi!
וצריך עיון גדול

These posts are intended to raise issues and stimulate further research and discussion on contemporary topics related to the daf. They are not intended as psak halacha.