as shown in fig1 , a hopper 10 rests on a surface 11 such as the ground . the hopper 10 is generally concave , having hopper walls 2 forming a chamber 1 with an open top 12 for receiving granular materials , such as grain , and directing the granular material downward along the incline of the walls 2 to a bottom 5 of the hopper 10 . for the removal of the granular materials , the hopper 10 is adapted to receive an inlet end 13 of an elongate screw conveyor or auger 14 . the auger 14 is adapted to extend from an inlet end 13 , residing adjacent a bottom 5 of the hopper 10 , and upwardly through the open top 12 at a front 8 of the hopper 10 , and extends therefrom for maximal evacuation of grain , typically directed at an elevated destination , such as a truck or a bin ( fig1 a ). with reference also to fig3 and 11 , a front wall 44 of the hopper 10 can distend outwardly for forming a protrusion 70 , the protrusion 70 forming an annular space 93 about the auger 14 for accommodating a circumferential guard 92 which typically extends circumferentially about the auger 14 . having reference to fig1 - 3 , the hopper 10 can have a generally oblique conical shape wherein the vertex or apex 3 of the oblique cone shape is oriented at the bottom 5 of the hopper 10 and a nominal center 7 of the open top 12 of the hopper is not aligned perpendicularly from the apex 3 . the open top 12 has an elongated shape . the nominal center 7 of the open top 12 is offset towards a forward end or front 8 of the hopper 10 . as a result , the incline of the walls 2 of the hopper from the front 8 to the apex 3 is less steep than is the incline of a back 9 of the hopper 10 to the apex 3 . as shown in fig2 - 6 , the hopper 10 has a composite shape . referring to fig2 and 6 , the hopper 10 is generally conical with side walls 2 diverging upwardly from the bottom 5 to the open top 12 . in embodiments of the invention , the hopper 10 disclosed herein has a configuration which deviates from a simplistic conical shape and other known profiles of conventional hoppers . with reference to fig1 , 3 , 4 and 5 the hopper 10 is supported on the ground 11 on a hopper base 21 ( fig1 a ) having a front - to - back profile b and a side - to - side profile c . in fig3 , viewed from the side and perpendicular to the longitudinal axis of the auger 14 ( removed for clarity ), the hopper 10 illustrates a narrow front - to - back base profile b for enabling rotation of the hopper 10 to adapt to the angle of the auger 14 . from the front view of fig4 , the hopper 10 has a relatively wide side - to - side profile c , providing lateral stability and to resist tipping . the hopper base 21 forms the bottom 5 of the hopper 10 and forms a chute for funneling granular materials to the auger 14 . with reference to fig6 , the side walls 2 of the hopper 10 are a merger of at least two major three - dimensional wall profiles ; a first three dimensional semi - conical profile 30 and a second three - dimensional ( 3 - d ) tetrahedron profile 40 . viewed in plan view ( fig2 and 11 ), the result of the merging of the profiles 30 , 40 is a chamber 1 having a generally elongate open top 12 , the nominal center 7 being offset to a front 8 of the hopper 10 . the first or semi - conical profile 30 is a cross - section of a right semi - circular conical portion 30 for forming back wall 31 of the hopper 10 having a forward facing and open front face 32 straddled by the back wall 31 . the conical portion 30 has a central axis x extending upwardly from about a bottom apex 3 . the second profile or tetrahedron profile 40 is a generally trigonal pyramid or tetrahedral - shaped portion 40 having substantially triangular side walls 44 forming a triangular v - trough 45 oriented forwardly . the side walls 44 of the v - trough 45 rise upwardly laterally and forwardly from a bottom edge 47 and diminish in height forwardly to the open top 12 , forming a prow p . the v - trough 45 has an open top 41 and a back - facing open back face 42 which is straddled by the front wall 44 . the front wall 44 of the v - trough 45 at the open back face 42 forms a wide back or stern which merges with the back wall 31 of the open front face 32 of the semi - circular back side 31 . a bottom apex 43 of the v - trough 45 is oriented at the bottom 5 . the v - trough 45 is truncated at the front to form a narrow front edge 46 at the open top 12 from which the auger 14 protrudes . preferably a semi - circular rest 50 is provided at the front edge 46 for conforming to and supporting the auger 14 at the open top 12 of the hopper 10 . the hopper front and back walls 44 , 31 at the merger of the open back and front faces 42 , 32 can include a third trapezoidal or substantially triangular or transitional profile or profiles 60 which merges the semi - circular 30 and tetrahedron 40 portions and forms connecting side walls 61 . the connecting walls 61 extend generally along a tangent from the back walls 31 , through the third substantially triangular profile 60 to intersect with the side walls 44 triangular v - trough 45 . the triangular v - trough 45 is widened through a fourth profile or protrusion 70 for increasing annular spacing 93 between the auger inlet end 13 and the hopper 10 . as shown in fig1 , the front wall 44 or v - trough 45 of the hopper 10 can distend outwardly and laterally forming the protrusion 70 which extends from about the hopper bottom 5 and partially along the front wall 44 for forming an annular space 93 about the auger 14 . the annular space accommodates the guard 92 and minimizes any incentive by an operator to remove the guard to clear more granular material . a fifth prism or wedge - shaped base portion 80 completes the bottom of the hopper structure and forms the hopper base 21 at the bottom 5 . the base portion 80 is narrow across base profile b to permit angular rotation of the hopper 10 and is wide along base profile c to stabilize the hopper 10 laterally . the base portion 80 has side walls 81 which intersect and merges with the semi - circular profile 30 and tetrahedral portion 40 . the side walls 81 have steep angle of inclination for funneling granular materials to the inlet end of the auger . as shown in fig7 , the auger 14 is a cylindrical tube 90 with a screw 91 extending coaxially within . the screw 91 protrudes from the cylindrical tube 90 at the inlet end 13 and is supported by a tail bearing 94 which is typically cantilevered from the cylindrical tube 90 . a circumferential cage or guard 92 surrounds the inlet end 13 between the tail bearing 94 and the tube 90 to exclude personnel from the screw 91 yet enable passage of granular material to the screw 91 . as shown in fig2 , 4 and 6 , the v - trough 45 along the edge 47 is widened to form the protrusion 70 which is a semi - circular or rectangular trough bottom having a nominal radius greater than that of the auger 14 so as to form the annular space 93 therebetween and aid grain flow into the inlet end 13 . the front wall 44 can be formed with one or more pairs of auger straddling slots therethrough for enabling cinch straps to pass around the auger tube 90 and be cinched or otherwise secured to the hopper 10 . each pair of slots can be spaced at differing angular positions for accommodating different sized augers 14 . fig7 - 15 illustrate a variety of views to show the relationship of the various features and configurations of the hopper 10 and how it adapts to accommodate the auger 14 . as shown in fig9 and 10 , the tip of the prow p is blunted or formed by the inclusion of the semi - circular rest 50 which is sized to support the cylindrical tube 90 of the auger 14 and maintain the annular space 93 in the protrusion 70 under the auger inlet 13 . best seen in fig1 a and 13 b , a stiffening lip 23 is formed about the open top 12 . the stiffening lip 23 is forgiving and will yield in the face of a point force or forceful deformation from impact and displacement , such as from encroachment of a truck box or other immovable structure thereon which might occur during operations such as unloading . the lip 23 is extends about substantially the entire open top 12 along an upper edge 22 of the side walls 2 formed by the semi - circular portion 30 , the tetrahedral portion 40 and the transitional portion 60 . the lip 23 first extends laterally outwards from the side walls 2 and then downwardly to a terminal edge . further , as best seen in fig1 b , circumferential ribs , or ribbed projections 24 , preferably of obtuse triangular cross - section , circumscribe a girth of the hopper 10 for imparting increased strength and structural stability to the side walls 2 . fig1 and 15 illustrate the inlet end 13 of the auger 14 with and without the guard 92 . the screw 91 has been removed to better illustrate the spatial relationship between the hopper 10 and the auger 14 . the tail bearing 94 is shown positioned in close proximity to the base portion 80 adjacent the bottom 15 . as shown in fig1 a and 16 b , the semi - circular back side portion 30 is amenable to easy access by the source of the granular material , such as by the tailgate of the grain truck 100 , from over about 270 degrees or at least three sides of the hopper 10 . the tetrahedron profile of the front of the hopper is strong . the aforementioned configurations of the hopper 10 enable low auger angles for maximal grain recovery and minimal residuals or loss when the hopper 10 is emptied . the combination of the semi - circular and tetrahedron portions 30 , 40 results in a strong hopper structure with a large volume or capacity . as shown in fig1 a - 17 b , the hopper 10 can pivot about the narrow transverse base profile b to conform the angle of the hopper side wall 2 to the angle of the auger 14 . the shallower the angle of the auger 14 , the more the hopper 10 will rotate to accommodate the auger 14 . as shown in fig1 , the hopper 10 is stackable in nested arrangement for minimizing shipping volumes with up to 20 % more hoppers 10 per shipment . for most applications , hopper diameters can be sized to about ½ of a maximal transport width enabling two - wide shipping arrangements for even greater shipping economies . further , the height of the open top 12 of the hopper 10 from the supporting surface or ground can be minimized for increasing the number of applications in which the hopper 10 can be used . the wide triangular v - trough 45 and protrusion 70 ensures that augers 14 can be used with the original guards 92 in place , thereby improving safety . long augers 14 , typically having angles of about 25 - 30 degrees are currently available and can be accommodated by rotation of the hopper 10 about the hopper base 21 .