Dofan Akuma and Stretching Walls of a Sukka to Connect to Distant Sekhakh

The mishna in Sukka (17a) cites several instances of a sukka whose walls are distant from the kosher sekhakh.  One example is a house which was converted into a sukka by opening a hole in the middle of the roof/ceiling and filling the area with sekhakh.  A similar case would involve a group of houses lining a courtyard with veranda-like extensions protruding from each unit.  If a person were to hang sekhakh from one veranda roof to another, he would effectively construct a sukka whose sekhakh is distant from the actual walls - in this instance, the walls of the houses which are separated from the sekhakh by the veranda overhang.  The mishna also cites a standard case of distant sekhakh, where a person fills the borders of the sekhakh roof with invalid sekhakh and inserts halakhically valid sekhakh only in the CENTER of the roof.  In each instance, the mishna validates the sukka as long as the distance between the valid sekhakh and the walls of the sukka does not exceed four amot.  In these cases, the principle of dofan akuma (literally, "a slanted wall") is employed, allowing the sekhakh to be placed at a less than four amah distance from the sekhakh.  This principle was transferred to Moshe as halakha le-Moshe mi-Sinai.
 

In two instances, Rashi (4a and 17a) interprets the principle of dofan akuma as a mechanism for converting invalid sekhakh located at the borders into an extension of the walls.  Although a typical wall is vertical, part of the wall can also run horizontally.  By viewing the base vertical wall as extending through the horizontally placed invalid sekhakh, the "slanted" wall actually DOES CONNECT with the kosher sekhakh located at the center of the roof.
 

The Ritva (4a) cites an alternative understanding of dofan akuma, which he believes is actually the position of Rashi (although it differs sharply with our version of Rashi). The Ritva claims that the principle of dofan akuma considers the wall of the sukka to be in the ACTUAL LOCATION of the valid sekhakh.  Instead of connecting with the pasul sekhakh and creating a wall at a right angle so that the actual wall connects with the valid sekhakh, dofan akuma simply allows the distant wall to be viewed as "connected" with the sekhakh even though there is no actual physical connection.  We imagine that the walls of the sukka slant forward to meet the kosher sekhakh located in the middle of the sukka.  From a geometric standpoint, employing dofan akuma converts a rectangular sukka into a trapezoid.
 

There are several differences which emerge from the opposing models of dofan akuma.  The most obvious question concerns sitting under the invalid sekhakh.  In Rashi’s classic version, in which the invalid sekhakh is considered part of the wall, one would not be permitted to sit under the invalid sekhakh. One is required to sit under sekhakh, not a wall, and the principle of dofan akuma transforms the invalid sekhakh into part of the wall.  The Ran explicitly claims that according to Rashi, the mitzva cannot be fulfilled by sitting under the invalid sekhakh that has now become a wall.
 

In contrast, the opposing model would allow for fulfilling the mitzva by sitting under invalid sekhakh.  According to this approach, the sekhakh has not become a wall; it remains sekhakh, but the wall is visualized as connected to the kosher sekhakh in the middle of the sukka.  By fashioning this trapezoid, we may mandate sitting anywhere in the sukka.  Anywhere you are positioned in the trapezoid, you are sitting under the upper parallel line.
 

A different question involves the application of dofan akuma in situations where the walls do not actually connect to the sekhakh.  The first model may only be applicable if the physical wall touches the sekhakh.  Only in this instance can we fashion a new right angle wall to connect with the kosher sekhakh situated at the center of the sukka.  If dofan akuma creates an imaginary slant of the outer walls to the kosher sekhakh, we would care little about the presence of a physical connection at the right angle corner between the vertical walls and the horizontal sekhakh.
 

An interesting related question is posed by the Korban Netanel, a commentary on the Rosh.  Rabbi Yoshia (Sukka 7b) claims that walls must also provide shade and therefore may not be constructed from transparent material. In explaining this position, which is not accepted as the halakha, several Rishonim assert that Rabbi Yoshia equated sekhakh and walls, applying all sekhakh standards to the walls themselves.  Would Rebbi Yoshia accept the notion of dofan akuma?  Would he allow converting invalid sekhakh into a wall and forming a right angle wall to connect to the kosher sekhakh in the middle of the sukka? The Korban Netanel claims that he would not, since anything invalid to be sekhakh is similarly invalid to be a wall.  Thus, the option of converting invalid sekhakh into a right angled wall is unacceptable.  Although the Korban Netanel’s position is debated, it is clear that if the dofan akuma principle transforms the rectangle into a trapezoid, even Rabbi Yoshia could accept it.  According to the second model, dofan akuma does not convert sekhakh into walls; rather, it migrates the wall in the direction of the kosher sekhakh.  This migration is possible even according to Rabbi Yoshia’s position, which applies sekhakh standards to walls.
 

Another interesting question involves applying dofan akuma to sekhakh which is higher than twenty amot.  The gemara (4a) describes a giant sukka to which was added a "stage," effectively raising the floor level and reducing the height of the sukka.  If this stage was built along one corner of the sukka adjacent to two walls, we may be forced to invoke dofan akuma to draw the third wall closer to the stage and provide a three walled sukka surrounding if (effectively yielding a three-walled sukka of less than twenty amot).  The gemara is initially unsure as to whether dofan akuma can help in this situation.  In its typical employment, dofan akuma moves walls to bypass/run through invalid sekhakh, but in this instance, the walls themselves are invalid because they are too high.  Can we use the principle of dofan akuma to move a wall toward a stage, thereby reducing its height and validating it for a sukka?  Can dofan akuma be employed to create halakhic walls or only to reconfigure the angles of a sukka and legitimate its dimensions? Despite initial doubts, the gemara ultimately concludes that dofan akuma can render halakhic walls as well.
 

Perhaps the deliberation surrounds the differing models of dofan akuma.  If the process recreates right angled walls, it might not be effective in reducing the height of the wall and creating a lower one.  The distance from the stage to the sekhakh that is at a right angle to the distant wall is indeed less than twenty amot.  But the part of the wall, including the sekhakh "converted" into wall, that spans the area beyond the partial stage is too high.  The magic of dofan akuma allows the walls to connect to the sekhakh above the stage and render that part less than twenty amot tall, but the remainder of the wall is still above twenty amot and therefore invalid.
 

If dofan akuma does not build right angle walls but rather trapezoid sukkas, it may indeed solve the problem of walls that are too high.  If dofan akuma stretches or slants the wall to meet the sekhakh above the stage, the top of the wall hits the sekhakh above the stage and no part of the wall is higher than twenty amot.  The gemara's deliberation as to whether to apply dofan akuma in this situation may be based on the gemara’s uncertainty as to the mechanism of dofan akuma.
 

This reading yields an interesting scenario.  It is possible that different SITUATIONS involve different versions of dofan akuma.  Merely connecting walls to distant sekhakh may only require the construction of right angled walls.  However, moving higher than twenty amot walls to meet the sekhakh above a constructed stage may require bending walls into a trapezoid form.  Dofan Akuma may be a halakhic principle which possesses multiple variations, some of which apply in one instance and others that apply in others contexts.  Asserting this variation may lead to different limitations upon dofan akuma in different applications.
 

Rav Soloveitchik suggested this approach to solve a contradiction in the Rambam.  When the Rambam describes the use of the dofan akuma principle to solve the problem of invalid sekhakh on the border, he defines dofan akuma as the construction of right angle walls.  When he describes dofan akuma as rendering lower than twenty amah walls, however, he speaks of bending walls into trapezoid form.  Perhaps the Rambam envisioned different models of dofan akuma operating under different circumstances.