Roofing tiles have dimensions differing from conventional tiles to allow manufacture of the tiles in a substantially greater size than conventional tile sizes. The tiles may be installed more rapidly, since each piece of material must be separately installed and a single tile may cover a greater area. Additional ribs are added underneath the interior portion of each tile than with conventional tiles. The ribs are spaced appropriately such that a person walking over each tile would have the weight of a single foot distributed over one or more ribs at all times. Moreover, the rims of each tile extend substantially deeper, through the thickness, of the tile. Each tile has material removed from the main surface portion, or the actual surface opposite the weather-exposed surface, to reduce the weight therefrom. Each tile may be configured with open air channels underneath each tile up and down the entire roof. Tiles may further be installed with no batten boards or furring strips, thus providing a complete availability of drainage and ventilation underneath the tiling system.

RELATED APPLICATIONS 
     This application claims priority to U.S. provisional patent application Ser. No. 60/077,003 filed Mar. 6, 1998. 
    
    
     BACKGROUND 
     1. The Field of the Invention 
     This invention relates to building products and, more particularly, to cast and extruded, cementitious tiles covering structures such as roofs. 
     2. The Background Art 
     Tiles have been used since ancient times. Clay tile is ubiquitous throughout Europe, the Americas, and other continents. Tiles produce many benefits. One of the benefits is longevity. Tiles, being manufactured predominantly of earthen materials, can survive the ravages of the elements. Nevertheless, tiles are heavy. Moreover, tiles can be rather fragile. High tensile strength is not normally available in tile materials. Moreover, adding the thickness of particular sections in order to improve strength properties becomes a very weighty proposition. 
     In modern construction, manufacturing processes, shipping, handling, breakage, installation, and so forth affect the utility of the materials. Lightweight is desirable, but unavailable in certain materials. Strength is a benefit, and is often relied on in materials, such as steel in place of wood, and so forth, in order to reduce weight while improving strength therein. 
     In roofing systems, asphalt shingles have been used for many years. In addition, other types of roofing based on manufacturing materials have been used. In addition, cedar shakes have been a preferred roofing material in certain environments. Nevertheless, wood being a plant material, inherently rots over time and decays, unlike earthen materials such as tiles. 
     Sealing a roof is a fundamental purpose of roof-covering materials. As a practical matter, a roof must have sufficient slope to shed rain, snow, and heat, effectively. A steeper pitch on a roof becomes problematic. Installation, maintenance, support, and the like for tiles may become a major issue. Thus, tiling systems are needed, which can provide sufficient structural integrity of tiles and which can be installed by methods that are sufficiently durable and economical. 
     Tiles may be walked upon by workmen during or after installation. Accordingly, breakage of tiles, especially near the overlap regions or in the center or unsupported region, is a common problem. 
     Breakage may expose, eventually, the interior of a building to water. Roofing systems must shed water and resist leaks. Roofing systems will typically support snow as it freezes, thaws, cycles through freezing and thawing, and eventually is melted or otherwise eliminated from a rooftop. 
     However, ventilation is not typically provided underneath a tile. Tiles typically close off the spaces underneath so that air is not able to flow upwardly or downwardly along the surface of a roof or otherwise underneath a tile. Moreover, condensation of humidity creates moisture underneath a tile. Wood strips, battens, cleats along the top edge of the tile, and other obstructions used in typical tiling systems may obstruct the flow of water resulting from the condensation. Accordingly, water cannot drain from underneath the tiling system. Also, a tile may break and produce a leakage path of moisture underneath the tile. Conventional tiling systems do not provide for ready runoff of such water. Thus, condensation, leakage, and ventilating air, are obstructed in conventional tiling systems. 
     What is needed is a tiling system for roofing that provides several advantages. A required advantage is lighter net weight of the roofing load. An additional advantage is greater strength for tiles in order to support against breakage by poor handling and walking on the roof by workmen. Also needed is a ventilation system for providing evaporation of any moisture that may accumulate beneath tiles in a roofing system, as well as providing drainage along the roof surface underneath the tiles. 
     Another need is a reduction of the damage produced by a tile system on the sealing material that may be placed over the fundamental structure of a roof. For example, rafters may support some kind of decking material, such as plywood or other sheathing. Over the sheathing may be placed a barrier, such as a vapor barrier, moisture barrier, or the like. For example, elastomeric polymer sheets may be used. Likewise, tar paper or asphalt roll paper, or felt, may be used. 
     Many sealing materials are available, but these materials are no match for the hardness, and abrasiveness of materials typically used in tiles. Accordingly, any tile resting on a surface covering may be cut through by tile edges with time, motion, and the presence of people walking thereon. 
     Thus, a tiling system is provided in accordance with the invention that obtains several structural advantages and advantages in installation. 
     BRIEF SUMMARY AND OBJECTS OF THE INVENTION 
     Roofing tiles made in accordance with the invention may be made by extrusion, casting, or other processes known in the art. The dimensions of the tiles are changed dramatically from those of conventional tiles. The tiles may be manufactured in a size that is substantially greater than conventional tile sizes. Accordingly, the tiles may be installed more rapidly, since each piece of material must be separately installed and a single tile may cover a greater area. Additional ribs are added underneath the interior portion, rather than around the border or edge of each tile. Moreover, the rims extend substantially deeper, through the thickness, of the tile. Material has been removed from the main surface portion, or the actual surface opposite the weather-exposed surface, to reduce the weight therefrom. However, the ribs are spaced appropriately such that a person walking over the tile would have the weight of a single foot distributed over one or more ribs at all times. 
     In the tiles made in one embodiment in accordance with the invention, longitudinal ribs and lateral ribs may both be provided. In addition, multiple longitudinal ribs and multiple lateral ribs may be provided. A lug or cleat may be provided for engaging a furring strip or batten. Nevertheless, the lugs may support the tile without resort to a batten or furring strip. Moreover, open air channels are maintained underneath each tile up and down the entire roof. Accordingly, in one presently preferred embodiment, tiles may be and should be installed with no batten boards or furring strips, thus providing a complete availability of drainage and ventilation underneath the tiling system. 
     The net thickness of the gutter section of each tile, engaging the next adjacent tile, is substantially thicker to greatly increase strength. For example, in most designs known in the art, engagement sections, keyed sections, overlaps and the like maintain less than half the net material dimension (transversely normal to the roof surface of the tile). These present less than an eighth of the nominal tile strength in the gutter area of the tile as opposed to the strength over the main area, for the engagement or overlap sections. In a design in accordance with the invention, the gutter thickness is substantially greater. Moreover, net width laterally is comparatively less. Since the strength is related to the third power of thickness, increasing the transverse dimension of any portion of the tile is substantially more effective than increasing the width in a longitudinal or lateral direction. 
     Thus, overlaps and ribs greatly increase strength, borrowing material from the thickness of clear spans therebetween. In one embodiment, a slanted edge or bottom surface of the ribs may be provided for fitting flat on a roof. This avoids any corners touching sealing materials or surfacing materials that may be placed underneath the tiles. Thus, the lower edge of a tile is ribbed, but each rib is angled to fit flat on the roof, while leaving a reinforced clear channel (for ventilation and drainage). Meanwhile, the top cleat at the top edge sits on the next tile up. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The foregoing and other objects and features of the present invention will become more fully apparent from the following description and appended claims, taken in conjunction with the accompanying drawings. Understanding that these drawings depict only typical embodiments of the invention and are, therefore, not to be considered limiting of its scope, the invention will be described with additional specificity and detail through use of the accompanying drawings in which: 
     FIG. 1 is top view of a tile in accordance with the invention; 
     FIG. 2 is a bottom view of the tile of FIG. 1; 
     FIG. 3 is a top view of an alternative embodiment of a tile; 
     FIG. 4 is a top view of an alternative embodiment of a tile; 
     FIG. 5 is a top view of an alternative embodiment of a tile; 
     FIG. 6 is a top view of an alternative embodiment of a tile; 
     FIG. 7 is a top view of an alternative embodiment of a tile; 
     FIG. 8 is a top view of an alternative embodiment of a tile; 
     FIG. 9 is a top view of an alternative embodiment of a tile; 
     FIG. 10 is a top view of an alternative embodiment of a tile; 
     FIG. 11 is a top view of an alternative embodiment of a tile; 
     FIG. 12 is a top view of an alternative embodiment of a tile; 
     FIG. 13 is a top view of an alternative embodiment of a tile; 
     FIG. 14 is a top view of an alternative embodiment of a tile; and 
     FIG. 15 is a top view of an alternative embodiment of a tile. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     It will be readily understood that the components of the present invention, as generally described and illustrated in the FIGS. 1 through 15 herein, could be arranged and designed in a wide variety of different configurations. Thus, the following more detailed description of the embodiments of the system and method of the present invention, as represented in the Figures, is not intended to limit the scope of the invention. The scope of the invention is as broad as claimed herein. The illustrations are merely representative of certain, presently preferred embodiments of the invention. Those presently preferred embodiments of the invention will be best understood by reference to the drawings wherein like parts are designated by like numerals throughout. 
     Referring to FIG. 1, a tile  10  may be constructed by one of a variety of processes, from a moldable material. In one presently preferred embodiment, a cementitious material, having a concrete-like appearance may be used. Suitable pigments, texturizers, strengtheners, and other materials may be formulated into a suitable composition. The material will preferably withstand the effects of weather by maintaining a comparatively impervious structure not susceptible to freezing, thawing, absorption of moisture, and the like. 
     In one embodiment, a tile  10  may be fabricated to extend in a longitudinal direction  11   a,  a lateral direction  11   b  and a transverse direction  11   c.  A tile  10  may be described in terms of a head  12  or head end  12  opposite a foot  14  or foot end  14 . A tile  10  may be installed to form a protective layer over a roof structure. The head end  12  is positioned vertically higher than the foot end along the slope of a roof. 
     In order to consistently shed moisture from rain, snow, and the like, a tile  10  may be formed to have laps  16 ,  18 . The sidelaps  16 ,  18  may be formed to lap or overlap one another on laterally adjacently positioned tiles. Also, tiles are installed in an overlapping arrangement of foot ends  14  resting upon head ends  12  of tiles  10  positioned adjacent one another in a longitudinal direction  11   a.  Thus, each foot  14  sits above and in contact with a head  12 , with respect to a transverse direction  11   c.  Similarly, tiles  10  installed adjacent one another in a lateral direction  11   b  overlap one another at respective sidelap portions  16 ,  18 . 
     Accordingly, a sidelap  18  or side overlap  18  will rest above (with respect to transverse direction  11   c ) a sidelap  16  of a laterally  11   b  adjacent tile  10 . Inasmuch as tiles  10  are installed on a sloped roof having a declining slope extending in a longitudinal direction  11   a,  a foot  14  may rest on a head  12 . However, since tiles  10  that are laterally  11   b  adjacent one another will have the same elevation at any particular point along a roof structure (underlying supporting structures such as battens, decking, etc.) the overlap  18  must fit over and within the underlap  16 . Thus, the underlap  16  may be thought of as a gutter portion  16 , while the overlap  18  may be thought of as a cover portion  18 . 
     In one presently preferred embodiment, a tile  10  may be designed to have a plurality of panels  20 ,  22 , rather than a single panel  20 . In conventional tiling an individual tile  10  has a single panel  20 , from which extend some form of underlap  16  and overlap  18  in a lateral direction. With a design made in accordance with the invention, two panels  20 ,  22  may be included in a single tile  10 , while having the appearance of comparatively large, conventional, individual, clay tiles  10 . 
     The panels  20 ,  22  may be delineated or bounded by a divider  24  or gutter  24  therebetween. The divider portion  24  may have a chamfer  25  similar, or identical, in appearance to the chamfer  25  provided on the underlap  16  and overlap  18 . The overlap section  18  may be provided with a matching bevel  26 . The bevel  26  also serves as a draft  26  for improving ease of release from a molding operation or molding machine. 
     For example, the tiles  10  may be made by extrusion, molding, and similar processes. A die may be used to shape the tile  10 , and particularly the Geometries of the panels  20 ,  22  and the laps  16 ,  18 . Accordingly, some amount of draft  26  may be appropriate. Moreover, structurally, a bevel  26  may provide additional structural integrity according to St. Venant's principal. That is, principal stress in isotropic materials typically acts at a forty five degree angle with respect to principal stresses (compression and tension). Accordingly, free rectangular corners are not helpful. Thus, a bevel  26  may be adapted to optimize weight, size, shape, and stresses within the tile  10 . 
     The bevel  26  may extend from a lower surface  27  to a ceiling  28  or finger span  28  cantilevered near the edge of the panel  20  forming a finger  30 . The finger  30  may angle downward to form a tip  32 , likewise angled or beveled somewhat like a mirror image of the bevel  26 , so far as the tip  32  extends. 
     The tip  32  terminates at a surface  34  or face  34  designed to interlock with the sidelap  16 . Thus, the tip  32  adjusts the position of the tile  10  with respect with to a sidelap  16  of an adjacent tile  10  laterally positioned next to an original tile  10 . 
     The finger  30  is adapted for strength and support superior to those produced by other methods in other apparatus. The strength of the finger  30  is increased substantially by redistributing the material in the tile  10  as described herein. Accordingly, the section modulus of the finger  30  is considerably greater than any corresponding appendage of a conventional tile, and yet is thinner than traditional clay, semi-circular tiles, traditionally seen for hundreds and thousands of years in Europe and the Mediterranean area. Moreover, the sidelap  16  may be designed in certain, presently preferred embodiments in order to completely underlie and support the entire sidelap  18 , even when staggering is used in installation processes or a staggered configuration is designed for offsetting the panels  20 ,  22  with respect to one another. 
     Referring to the sidelap  16 , also referred to as the gutter portion  16  of the tile  10 , a bevel  36  or draft  36  may extend from a chamfer  25  off the top surface  37 , downwardly to a gutter surface  38  or face  38  for transporting moisture longitudinally  11   a  along the tile  10 . The gutter surface  38  or surface  38  may be thought of as forming a channel  46  interweaved with a channel  48  of the sidelap  18 . One may note that each of the channels  46 ,  48  need only conduct moisture away from the individual tile  10  to which they pertain, since moisture is discharged onto a longitudinally  11   a  adjacent tile  10  therebelow. Likewise, a comparatively close fit between the face  34  and the face  38  need present no difficulty to the transport of moisture. Capillary action is operational and of substantial effect in gaps less than approximately one quarter inch. Accordingly, surface tension forces can maintain moisture against overflowing the gutter  46 . 
     The sidelap  16  underlying the sidelap  18  may have a toe  40  for extending under the sidelap  18  across the surface  34 . The toe  40  may terminate with a tip  42  provided with a face  44  or surface  44  adapt to fit within the channel  48  against the surface  28  of the sidelap  18 . 
     In one embodiment, the channels  46 ,  48  may be very nearly mirror images of one another, particularly insofar as the finger  30  and toe  40  are concerned and with respect to the tip  32  and tip  42 . Nevertheless, the bevel  25  may provide a practical benefit in relieving mold surfaces for providing easy removal of a tile  10  during molding or extrusion processes. Since no particular benefit would be gained by placing a bevel  25  on the tip  42  near the bottom surface  27 , exact, mirror identity is not necessarily required or useful between the sidelap  16  and the sidelap  18 . Thus, a gutter channel  46  is adapted for underlying a cover channel  48 . The finger  30  and tip  32  may thus engage the toe  40  and tip  42  in an alignment function, as well as a sealing function. 
     Tiles  10  may be attached to a roof structure in one of a variety of ways. Typically, tiles may be formed with a cleat near an upper end  12  or head  12  for engaging a purlin or a batten extending between rafters of a roof truss or attached to decking over rafters of a roof structure. In accordance with modern earthquake and storm provisions of building codes, a tile  10  may be adapt to be secured to a roof structure. Accordingly, apertures  49  may be disposed, periodically or otherwise, near the head  12  of a tile  10 . Each aperture  49  may be sized to receive a nail or fastener of suitable dimension to secure the tile  10  against a roof structure. 
     Because the tiles  10  do not require battens, furring strips, or purlins in order to engage a roof structure, the apertures  49  may be provided differently from conventional tile systems. For example, more apertures  49  may be provided. Traditionally, not every traditional tile is securely fastened to a roof structure in a transverse direction  11   c.  However, having several apertures  49  in each tile  10  in accordance with the invention may provide for fastening all tiles  10  exclusively by fasteners throughout the apertures  49 . Accordingly, screws, available in automated equipment, nails, staples, studs, pegs, brackets, and the like may extend throughout the apertures  49 . 
     In one embodiment, a tile  10  may be lifted at the foot  14  to expose a longitudinally  11   a  adjacent tile  10  for replacement. In such a case, a short cleat or batten strip may be placed under the tile  10 , obviating, for a single, new, replacement tile  10  a need to reuse nails and apertures  49  in an inaccessible location. Nevertheless, tiles  10  may be installed exclusively with fasteners throughout the apertures  49 , without the need for battens extending laterally  11   b  across the entire roof structure. 
     A tile  10  may have a maximum thickness  50  extending between a bottom surface  27  and a top surface  37 . Nevertheless, the thickness  50  in a tile  10  in accordance with the invention is designed to provides substantial improvements over conventional tiling systems. For example, the thickness  50  may be substantially greater than that of conventional tiles, while not being uniform over the entire tile  10  in any dimension  11   a,    11   b.  Notwithstanding the sidelaps  16 ,  18 , the thickness  50  may be substantially greater than a corresponding thickness of a conventional tile while redistributing material in a tile between the head  12  and foot  14  and between the sidelaps  16 ,  18 , as will be discussed hereinafter. 
     A depth  52  of a channel  48  may be provided to clear the surface  44  and tip  42  of the toe  40 . Meanwhile, the depth  54  of a chamfer  25  may be selected for aesthetic reasons, as well as to provide a substantially clear and guiding conduit for moisture running off a tile  10 . 
     The chamfer  25  is not required. Nevertheless, the chamfer  25  also provides a certain amount of relief in placing tiles, while relaxing tolerance requirements and providing for an easier fit and less breakage in fitting tiles  10  together. 
     A depth  56  of the tip  32  of the finger  30  may be designed to match the tip  42  of the toe  40 . Accordingly, a clearance  58  is provided for accommodating the toe  40  and surface  38 . In one embodiment of an apparatus  10  in accordance with the invention, the clearance  58  may be substantially greater, due to the additional material and size provided for the toe  40  as well as the finger  30 . 
     For example, the thickness  60  of the finger  30  may be substantially greater than would be a corresponding thickness in a conventional tile. A section modulus relates directly to maximum stress at an outermost fiber in a flexurally loaded (bending) beam. Moreover, a section modulus is proportional to a base or width to a first power, and a depth  60  to a third power. Accordingly, any improvement in a depth  60  of a finger  30  is compounded by a third power. For example, doubling the thickness  60  will multiply a section modulus by a factor of 8. Thus, the depth  56  of the tip  32  may be comparatively shorter, in order to accommodate a greater thickness  60  in the finger  30 , as well as increasing the dimension of the toe  40 . 
     Referring to FIG. 2, and generally to FIGS. 1-4, a depth  62  of a gutter channel  46 , may be sized to accommodate the finger  30  and finger tip  32  of the sidelap  18 . Accordingly, a toe height  64  may selected for fitting within the cover channel  48 . A tip height  66  may be adapted to matingly engage the tip  32  of the channel  48 , interlocking the sidelaps  16 ,  18  of adjacent tiles  10 . In one embodiment, the thickness  70  of the toe  40  may be exactly the same as the thickness  60  of the finger  30 . 
     Nevertheless, the exact dimensionalities need not be equal. Rather, corresponding and mating dimensions of the finger  30  and toe  40  need to be consistent with one another. Also, to provide adequate strength for supporting service, walking over by persons, installation, and handling abuses, maximum thicknesses  50 ,  60 ,  70  are desirable. Nevertheless, the channel  46  must carry runoff moisture. Therefore, the height  66  of the tip  42  should be sufficient that any head height of moisture in the channel  46  may be overcome by the height  66  and the surface tension of the moisture combined. 
     Referring to FIG. 1 again, a tile  10  may have a length  72  selected to meet several criteria. Similarly, a width  74  may be selected in a lateral direction  11   b  in accordance with various, applicable, appropriate criteria. For example, building codes require tiles  10  to maintain sufficient strength to support a person walking thereon. Thus, a tile  10  made in accordance with the invention needs to have sufficient strength, alone, to survive testing adapted to determine the ability of a tile  10  to support a person walking thereon. Also, a length  72  may be adapted to provide an aesthetic appeal of the amount of each panel  20 ,  22  exposed when longitudinally  11   a  adjacent tiles  10  are overlapped. 
     Also, a length  72  may be sufficiently long to minimize labor involved during installation. For example, a longer length  72  implies that fewer tiles  10  may be required to cover a particular expanse in a longitudinal direction  11   a.  Also, the length  72  may contribute to an increased moment due to loading near the center of a tile  10 . Accordingly, an excessive length  72  begins to increase the maximum load in the outermost fiber (maximum dimension from a neutral axis, from an engineering perspective) of the tile  10 . 
     In one embodiment, a suitable length  72  may also be selected to optimize the total weight of a tile  10 . For example, an installer must lift each tile  10  from a stack and position the tile  10  on a roof structure for installation. Excessively large dimensions  72 ,  74  may result in too much weight in each individual tile  10 , or merely a very unwieldy size, inconvenient and tending to increase breakage from dropping of striking other objects. 
     In one embodiment, a length  72  may be approximately 16 inches. In an apparatus  10  in accordance with the invention, a length  72  of 16 inches provides superior strength over conventional tiling systems. Similarly, the width  74  may be selected according to similar criteria, including, support, handling, and the like. However, conventional tiles do not provide a width  74  corresponding to the width  74  of the tile  10 . The width  74  may be approximately 16 inches also. 
     Increased width  74  has an advantage in that the loss of coverage due to overlapping of the sidelaps  16 ,  18  in adjacent tiles  10  becomes a lower fraction of the overall dimension  74 . Accordingly, more coverage may be obtained from fewer tiles  10 . Moreover, a tile  10  in accordance with the invention relies on a substantial redistribution of material throughout the tile  10 , in order to optimize strength and weight. Because the material has been redistributed, as compared with conventional tiles, a tile  10  may have increased strength, and reduced weight over tiles made by conventional methods and having lesser dimensions  72 ,  74 . Thus, the net effect of a tile  10  in accordance with the invention is a lighter tile  10 , covering more roof area with a fewer number of tiles, with superior strength and resistance to breakage due to mishandling and walking thereon by people. 
     Referring to FIG.  2  and continuing to refer generally to FIGS. 1-4, a width  76  of a finger span  28  affects strength of the finger  30  directly, to a first power. By contrast, the thickness  60  of the finger  30  affects strength to a third power. The same principals apply to the comparative widths  78 ,  80  of the tip  32  and lap  18 , respectively, with respect to the depths  60 ,  56 . Thus, a tile  10  in accordance with the invention may rely on a net width  80  of a lap  18  substantially less than a corresponding overlap of a conventional tile. The width  80 , due to the mating nature of the sidelaps  16 ,  18 , corresponds to the same width of the sidelap  16  forming the gutter channel  46 . 
     A panel  82 , of which the panels  84 ,  86  may be considered instances, may extend across portions of the tile  10 . Material is exchanged from the region of the panels  82  in order to provide reinforcement selectively about the tile  10 . Such redistribution provides additional strength as needed in the tile  10 . 
     However, lugs  88  or cleats  88  may be provided, extending from the under surface  27  of a tile  10 . The lugs  88  may be positioned to support the tile  10  between the head  12  resting on a roof structure, and the foot  14  resting on a longitudinally  11   a  adjacent tile  10 . 
     The cleats  88  or lugs  88  may be tapered along a longitudinal angle  89   a  and a latitudinal angle  89   b.  The longitudinal angle  89   a  and the latitudinal angle  89   b  may be unique to the lugs  88 , but may also be the same for ribs  90 ,  92 ,  94 . The ribs  90 ,  92 ,  94  extend across the bottom surface and may be disposed longitudinally, laterally, as well as diagonally. The ribs  90 ,  92 ,  94  may be disposed in various angles with respect to the head end  12  and the sidelap  16  to thereby increase a strength to weight ratio of the panel. 
     In one presently preferred embodiment, the ribs  90  extend laterally  11   b  across the tile  10 . Each has a corresponding height  91   a  extending away from the panels  82 , a base width  91   b  extending longitudinally  11   a  along the tile  10  at a panel  82 , and may have a top width  91   c  extending longitudinally  11   a.  Nevertheless, in one presently preferred embodiment, each of the lateral ribs  90 , of which the ribs  92  are a particular instance, may be rounded for structural, aesthetic, material, and airflow reasons. 
     The ribs  94  extend in a longitudinal direction  11   a.  The ribs  94  may be tapered at an appropriate lateral angle  89   b.  Similarly, the lateral ribs  90  may be tapered at a longitudinal angle  89   a.  The angles  89   a,    89   b  need not be identical for the lugs  88  and ribs  90  or ribs  94 . 
     The panels  82 , of which the panels  84 ,  86  are specific instances positioned longitudinally  11   a  toward the head  12 , may contain less material than would a corresponding region in a conventional tile. The material that could have been applied to the panels  82  between the top surface  37  and bottom surface  27  of the tile  10  is instead distributed through the ribs  90 ,  96 . Material removed from the panels  82  or redistributed from the panels  82 , reduces the flexural modulus or section modulus of each panel  82  over a comparatively large area. Redistributed to the ribs  90 ,  96 , the material increases substantially the section modulus of the ribs  90 ,  96 . 
     Each of the panels  82  can well afford decreased strength when supported by underlying ribs  90 ,  96 . Again, since section modulus is proportional to a third power of a depth dimension, ribs  90 ,  96  substantially increase the section modulus of the tile  10 , while the comparatively small reduction in the section modulus of the panels  82  provides an inordinately greater modulus in the ribs  90 ,  96 . Moreover, the panels  82  do not have extensive clear spans. For example, no panel  82  is left unsupported in a lateral direction  11   b  across the entire width  74  of the tile  10 , as would be the case in conventional tiles. Thus, the redistribution provides a minimal reduction in section modulus for the panels  82 , which do not need strength, being reinforced by ribs  90 ,  96 , and provides great increases in the section modulus of the tile  10  overall by way of the ribs  90 , 96 . 
     In one embodiment, each of the ribs  94 , may be tapered through an angle  95  near the head  12  thereof The angle  95  may be selected to correspond to the angle at which each tile  10  is disposed when resting at the head  12  on a roof structure, and at the foot  14  on a longitudinally  11   a  adjacent tile  10 . 
     The angle  95  may also be provided as a modification of each of the cleats  88  or lugs  88 . Thus, greater surface contact area on a supporting roof structure is available to the lugs  88  and ribs  94 . Meanwhile, since each tile  10  will be disposed at this same angle  95 , along a top surface  37 , parallel to all other tiles  10  in the roof system, the foot  14  may be supported substantially along the entire width  74  thereof by a flat upper surface  37  of a longitudinally  11   a  adjacent tile  10  therebelow. 
     Eliminating corners from the ribs  94  during contact with an underlying roofing structure distributes the weight of tiles  10  over a substantially larger area than would conventional tiles. Moreover, since the lugs  88  also provide substantial surface area in flat contact with the underlying roofing structure, any sealing materials placed over a roof structure and beneath a tile  10 , will receive substantially lower stresses, reducing leakage, and the possibility of tearing through. 
     The ribs  96  are instances of ribs  94 , disposed near the sidelaps  16 , 18 . As with all the longitudinal ribs  94 , the tapered portions  97  are adapted to accommodate the installation angle  95  of tiles  10 . Since the depth  70  or thickness  70  of the toe  40  is increased over that of conventional tiles, the outer ribs  96  have sufficient dimensions  70  to accommodate the tapered sections  97 . 
     In one embodiment, a tile  10  may be provided with an additional bulkhead  98 . The bulkhead  98  may serve as a lateral rib  90 . Nevertheless, the bulkhead  98  may also serve as a foot  14  of one panel  22  of a tile  10 . For example, a die may be manufactured to provide the shape of the panels  82 , lugs  88 , and ribs  90 ,  94 . An adaptation may provide a core to fill in the portion of the foot  14  beyond the bulkhead  98 . Accordingly, the bulkhead  98  and extension  99  may become the outermost boundaries, in their respective regions, of the tile  10 , and the panel  22 , in particular. The shortened appearance of the bulkhead  98 , positioned away from the foot  14 , provides an additional appearance of multiple tiles of conventional dimensions laid in a staggered or offset pattern for aesthetic appeal. When a core is removed from a die, the foot  14  may extend straight across  116  the entire width  74  of the tile  10 . 
     The panels  82  may be thought of as having a consistent and uniform dimension in a transverse direction  11   c.  Thus, each of the panels  82  may be thought of as comprising a plate  100  or as forming a portion of a plate region  100 . As a practical matter, the material in the ribs  90 ,  94  as well as in the lugs  88 , is integrally contiguous, and molded in a single cavity of a die. Nevertheless, the thickness  101  of a plate portion  100  may be considered a new nominal thickness  101  of a tile  10 . 
     Alternatively, the thickness  50  may be considered a nominal thickness  50  of the tile  10 . However, one may see that the thickness  50  becomes a thickness  50  corresponding to ribs  94 , while the nominal thickness  101  is a thickness  101  of plates  100  extending over the tile  10  as panels  82 . Each of the plates  100  or panels  82  has a corresponding span  102  or clear span  102  extending in a lateral direction  11   b.  The width  11   b,  unsupported between the ribs  94  is the maximum dimension that any plate  100  must support. Accordingly, the thickness  101  may reduce without risk of failure during tests or service. In one embodiment, the width  102  between ribs  94  is selected such that a shoe of a person walking on the top surface  37  of a tile  10  will always be supported by at least one rib  94 . 
     Due to the lateral angle  89   b  of taper in each of the ribs  94 , a width  103  or base width  103  of each rib  94  may be greater than the width  104  or face width  104 . The angles  89   a,    89   b  accommodate draft for molds, while optimizing structural strength at minimum weight in each of the ribs  94 . Thus, at a height  105  or depth  105  of a rib  94 , the width  104  need not be as great as the base width  103 . As a practical matter, the lateral angle  89   b  for the draft on any rib  94  may be a 45 degree angle. However, maximum stress is allowable for compression, but this need not be a limiting factor. Therefore, in order to extend a maximum depth  105  below the plates  82 ,  100 , the ribs  94  may have a substantially steeper angle  89   b,  closer to a right angle with respect to the lower surface  27 . 
     The height  106  or depth  106  corresponding to the declination length  107  and declination depth  108  of the tapered section  97  of ribs  94  may be selected to maintain the integrity of each of the channels  46 ,  48 . 
     The span  110  extending longitudinally between adjacent lateral ribs  90 ,  92  may be selected in accordance with required strength for a maximum unsupported distance in the tile  10 . However, the grid work of ribs  90 ,  94 , in conjunction with the panels  82 , “box in” the undersurface  27  of the tile  10 , greatly increasing the strengths thereof. The spans  110 ,  112  need not be equal. For example, overlapping heads  12  and feet  14  in the tiles  10  provide support. Likewise, the lugs  88  provide support. Finally, under the head  12 , the tapered portions  97  of the ribs  94  provide support. Thus, the portions of the overall length  72  distributed between the ribs  90 ,  92  and the head  14  may be optimized for equalizing stress, and minimizing stress throughout the tile  10 . 
     The span  113  is optional. For example, the bulkhead  98  need not be present. Nevertheless, the extension length  113  may be provided as the center-to-center distance between the ribs  90 ,  92 ,  98 , just as the spans  110 ,  112  may. 
     Similarly, a span  114  represents a center-to-center distance, if the ribs  94  are uniformly spaced. Thus, a center-to-center distance  114  may be measured between any corresponding points on adjacent ribs  94 . The span  116 , may represent the clear span  116  between the ribs  94 . Nevertheless, with tapered ribs  94 , the net clear span  102  is the maximum span actually clear and unsupported in the panels  82 . 
     Moreover, since the panels  82  are integrally formed with the ribs  90 ,  94 , the span  102  is not simply supported. Therefore, additional strengths are available in the panel  82  above that of a simply supported plate of the same dimension. Meanwhile, the span  118  in a longitudinal direction  11   a  for each plate  82  may be designed in conjunction with the ribs  94 . Accordingly, the clear span  118  and the clear span  102  may provide proper dimensions to effectively distribute weight and distribute stress more uniformly within a tile  10 . 
     One may see that comparatively larger sizes  72 ,  74  of tiles  10  may be manufactured. Additional cleats  88  or lugs  88  may be added. Lugs  88  may be distributed along the longitudinal direction  11   a.  Similarly, depth  105  and widths  103 ,  104  of ribs  94  may be adapted for a particular configuration of clear spans  102 ,  118  in the panels  82 . 
     Supplementary, lateral ribs  92  may be distributed longitudinally  11   a  to extend laterally  11   b  between the ribs  94 . Thus, one may see that a tile  10 , made in accordance with the invention, obtains superior strength, reduced stress, and lighter total weight than conventional tiles. Redistribution of material from the panels  82  or plates  100  into the ribs  90 ,  94  provides substantial flexibility of design in selecting a length  72  and width  74  of a tile  10 . Moreover, provision of lugs  88  adapted to particular positions along the length  72  may decrease the net unsupported length of any rib  94 . In fact, lugs  88  may be added on every rib  94 . However, in certain embodiments illustrated, current building code requirements can be met, cost and weight can be reduced, while substantial labor is saved. 
     Referring to FIG. 3, while continuing to refer to FIGS. 1-4, a face  120  at the foot  14  of a tile  10  is typically visible to a passerby viewing a tiled roof. Accordingly, the face  120  may be fluted, scalloped, notched, beveled along a transverse direction  11   c,  and so forth. Accordingly, architectural, stylistic preferences may be accommodated by the treatment of the face  120 . 
     Moreover, a face  121  may be offset in a longitudinal direction  11   a  away from the face  120 . Thus, a setback  122  or length  122  of setback between the face  121  and the face  120  may be provided. The effect of a setback  122  is to give the appearance of multiple tiles when the panels  20 ,  22  are viewed for a single tile  10 . Moreover, alignment by a workman of multiple tiles, or rather misalignment in order to provide the offset  122 , may be unnecessary. Thus, work is sped up for installers. 
     The foot  14 , is thus augmented by a shortened foot  124  corresponding to the face  121 . However, a principal function of a tile  10  is protection of a roof against the elements. In order to operate effectively, a tile  10  must be robust for installation and service. A person walking on the foot end  14  of a sidelap  18  that is unsupported may break the cantilevered finger  30 . Therefore, in one embodiment of a tile  10  in accordance with the invention, a spur  126  may be provided on the sidelap  16  forming the gutter channel  46 . 
     The spur  126  may extend the setback distance  122  from the face  121 . Accordingly, the spur  126 , when installed, lies flat, parallel to a top surface  37  of a longitudinally  11   a  adjacent tile  10 . Thus, the spur  126  is completely supported in compression underneath a sidelap  18  engaged therewith. Thus, a finger  30  extending over and into the spur  126  is completely supported along the entire length  72  of a panel  20 . Therefore, even though a panel  22  may be shorter than a panel  20  having a length  128  shortened by the setback  122 , no portion of the tile  10  is left unsupported by the roof structure. 
     For aesthetic reasons, the faces  120 ,  121  may be beveled. A bevel angle  129  or bevel  129  has several effects. First, the setback  122  need not be as extreme to achieve the same apparent offset  122  when viewed visually. Meanwhile, the underlying surface  27  under the foot  124  may be extended longitudinally  11   a  substantially closer to the foot  14 , while giving a maximum appearance of offset  122 . Moreover, since the angle  129  affects the appearance of light and shadow reflected from the surfaces  37  and faces  120 ,  121  a “difference” exists between any light reflected from the surfaces  120 ,  121 . This difference provides an appearance of distinct surfaces  120 ,  121 , accentuating the influence of the offset  122 . Meanwhile, the underlying surfaces  27  at the foot  124  may provide more structural overlap over a head  12  of a longitudinally adjacent tile  10 , improving strength while maximizing the coverage of tiles  10  with a minimum number of tiles  10 . 
     One may think of the face  120  of the foot  14  corresponding to a long overlap section  130 , with the foot  124  and surface  121  corresponding to a short overlap section  132 . The bevel angle  129  maximizes the coverage by tiles  10  while maximizing the appearance of variation due to offsets  122  while providing mechanical strength according to St. Venant's principal of stress distribution along principal stress lines. Thus, the bevel angle  129  may angle back at up to forty-five degrees while maintaining substantially the same strength. Thus, the angle  129  may be limited only by aesthetic considerations, and stress limitations. 
     Referring to FIG. 4, and continuing to refer generally to FIGS. 1-4, an alternative embodiment of an offset  122  may be used. Unsupported, cantilevered sidelaps  18  may be broken more easily by the weight of a shoe of a person walking thereon. Such a problem may be exacerbated by the rise between the tip  32  and the head  12  of the next, lower, longitudinally  11   a  adjacent tile  10 . Thus, a person stepping on the sidelap  18  near the foot  14  may break the finger  30 . The reason that the finger  30  may be unsupported near the foot  14  is the very reason why breakage must not occur. The unsupported finger  30  is the only roof protection against moisture. Accordingly, a broken corner of a tile  10  may immediately produce a leak directly under a channel  46  of a tile  10 . Thus, resistance to breakage may be a significant need. 
     In one embodiment, a notch  134  lacks the spur  126  of FIG.  3 . Instead, the full width  135  of a panel  22  including the sidelaps  16  forming the gutter channel  46 , is clear between the face  120  and the face  121 . Correspondingly, a notch  136  may be formed in the panel  20 . The notch  136  exposes the long overlap  130  extending to the face  120 . The long overlap  130  and the short overlap  132 , after installation, rest on the head  12  of a downwardly, longitudinally  11   a  adjacent tile  10 . Thus, the setback  122  may be common (have a common value) for the face  121  and the notch  136 . 
     Nevertheless, the offsets  122  at the notch  136  and the notch  134  need not be exactly equal. However, in one presently preferred embodiment, the offsets  122  may be identical for the notch  134  and the notch  136 . Thus, a clean appearance is obtained in which the face  121  terminates longitudinally  11   a  with the notch  136 , exposing the long overlap  130 . Meanwhile, the entire sidelap  18  or finger  30  is completely supported by an underlying, fitted, toe  40 . The tip  42  fits into the channel  48  completely supporting the finger  30 . The tip  32  is similarly supported by the toe  40  in the gutter  46 . 
     A tile  10  does not typically have a substantial amount of strain available. Cementitious materials tend to have greater strength in compression than in tension. Nevertheless, a tile  10  made in accordance with the invention has sufficient strength to greatly improve the durability after installation. Moreover, the tiles  10  are not rigidly connected. Rather, fasteners throughout the apertures  49  prevent sliding of tiles  10  longitudinally  11   a  down a roof structure. Meanwhile, underneath a headlap  138  or head endlap  138 , the tapered portions  97  of the ribs  94  present flat surfaces  140  parallel to the underlying roof structure. Similarly, the lugs  88  present contact surfaces  142  distributing their loads flat against the underlying roof surface. Therefore, virtually every portion of the tile  10  is fully supported by direct contact of materials compressed therebelow or by the lattice work of ribs  90 ,  94  and supporting lugs  88  over the spans that would have otherwise been unsupported. 
     One advantage of the embodiment of FIG. 4 as compared with the embodiment of FIG. 3 is that the spur  126  need not be present. Although equally durable in most situations after installation, the spur  126  may present greater handling difficulties, becoming susceptible to breakage as a lone cantilevered part  126 . 
     From the above discussion, it will be appreciated that the present invention provides several new features. Special and unique attributes of a new interlocking concrete roof tile design include tiles that are larger in size, (e.g. 16″×16″) to maximize the structural flexural strength requirements of the Uniform Building Code, without exceeding the maximum standard allowable width. Midpoint support and batten engagement lugs reduce the structural span to increase dramatically the actual flexural strength performance. This also allows the roof deck load bearing to be distributed over 8 strength ribs  90 ,  94  and lugs  88  instead of one or two lugs  88  as with common tile. This tile  10  is designed to be installed directly to roof deck without battens, and the unique design allows moisture and air to move freely upslope under the tile  10 , without the obstruction related to standard tile lug designs, producing a legitimate “cold roof” system. 
     High strength structural ribs may run longitudinally through the tile. Several (e.g. three) of the ribs  90 ,  94  are modified in one embodiment to include a midspan support and batten engagement lug  88 , which bear directly on the roof deck. This maximizes and multiplies the flexural strength across the unsupported span of the tile. The horizontal strength rib at the top of the tile  10  has been designed to allow underlayment drainage in a straight, unobstructed pathway to the eave of the roof, and this also allows upslope air movement from eave to ridge (e.g. a cold roof system) that reduces ice buildup in the winter and attic temperatures in the summer. 
     The tile is designed to be efficient to install. It uses, in one embodiment only 71 tile per square (100 square feet). Prior art tile products use 89 to 150 pieces per square. This means fewer nails to fasten, and fewer tiles to handle. Since designed to install directly onto the deck without the use of battens on slopes below {fraction (7/12)}, it reduces labor and costly accessory products. The “cold roof” qualities of the invention reduce labor by 40% as compared to prior art tile “cold roof” systems, requiring vertical and horizontal strips to accomplish the same benefits. 
     When tested according to I.C.B.O. and U.B.C. requirements the new tile design yields a substantial improvement in strength. The unique lateral strength rib is placed where it is most able to provide strength to resist breakage from foot traffic and hail damage. The midspan support and batten engagement lugs are strategically placed for maximum strength and minimum installed weight. A “staggered” appearance with various types of “rustic” rough cut edges may be presented without actually staggering the tile installation. 
     The tile may incorporate a feature of marking layout to position the headlap of the tile. Marks showing the proper overlap distance may be marked directly on the tiles to aid the installer to properly position the tile with the required headlap for coursing layout from eave to ridge. A taper allows the tile to distribute its weight onto several (e.g. five) broad support areas and minimize the thickness at the upslope lap, such that the slope loss of the tile in installation, is minimized. 
     The tile  10  shown in the embodiments of FIGS. 1 through 4 may be embodied to have various types of top surfaces  37  and foot ends  14 . In the following FIGS. 5 through 15, various possible embodiments are shown. One of skill in the art will appreciate that additional embodiments which combine various features are possible and are included within the scope of the invention. 
     Referring to FIG. 5, a top view of an alternative tile  10  is shown. The tile has two panels  151 ,  152  which are divided by a divider  24  similar to previous embodiments. Panel  151  is referred to herein as the lapped panel due to its lower sidelap  16 . Panel  152  is referred to herein as the lapping panel due to its upper sidelap  18 . The face  120  illustrates the profile  153  of the lapped panel  151  and the profile  154  of the lapping panel  152 . The panels  151 ,  152  have corresponding top surfaces  155 ,  156 . 
     The embodiment of FIG. 5 is similar in most respects to that of FIG.  1 . The primary difference is that the top surfaces  155 ,  156  comprise an undulating structure  157 . The undulating structure  157  provides a rough textural appearance which serves for aesthetic purposes. 
     Referring to FIG. 6, a top view of an alternative tile  10  is shown and comprises a lapped panel  161  and a lapping panel  162 . The lapped panel  161  and the lapping panel  162  have corresponding profiles  163 ,  164  and top surfaces  165 ,  166 . The top surfaces  165 ,  166  are smooth or generally flat in this embodiment. The embodiment of FIG. 6 is primarily that of FIG. 4 except that the panels  161 ,  162  have jagged, oblique edges  167 ,  168  as indicated. The exact nature of the jagged edges may differ as desired as such edges serve an aesthetic purpose. 
     Referring to FIG. 7, a top view of an alternative tile  10  is shown and comprises a lapped panel  171  and a lapping panel  172 . The lapped panel  171  and the lapping panel  172  have corresponding profiles  173 ,  174  and top surfaces  175 ,  176 . The embodiment of FIG. 7 is similar to the embodiment of FIG. 4 except that the embodiment of FIG. 7 has an undulating structure  177  on the top surfaces  175 ,  176 . The undulating structure  177  is similar to that of the undulating structure  157  of FIG.  5 . The undulating structure  177  continues along the length of the overlap portion  178  of the lapping panel  172 . 
     Referring to FIG. 8, a top view of an alternative tile  10  is shown and comprises a lapped panel  181  and a lapping panel  182 . The lapped panel  181  and the lapping panel  182  have corresponding profiles  183 ,  184  and top surfaces  185 ,  186 . The top surfaces  185 ,  186  may be characterized as flat or generally smooth in this embodiment. The embodiment of FIG. 8 is similar to the embodiment of FIG. 3 except that the embodiment of FIG. 8 has a beveled edge  187  on the lapped panel  181 . Furthermore, the lapping panel  182  has a jagged, oblique edge  188 . 
     Referring to FIG. 9, a top view of an alternative tile  10  is shown and comprises a lapped panel  191  and a lapping panel  190 . The lapped panel  191  and the lapping panel  192  have corresponding profiles  193 ,  194  and top surfaces  195 ,  196 . The top surfaces  195 ,  196  may be characterized as flat or generally smooth in this embodiment. The embodiment of FIG. 9 is similar to the embodiment of FIG. 3 except that the panels  191 ,  192  have jagged, oblique edges  197 ,  198 . 
     Referring to FIG. 10, a top view of an alternative tile  10  is shown and comprises a lapped panel  201  and a lapping panel  202 . The lapped panel  201  and the lapping panel  202  have corresponding profiles  203 ,  204  and top surfaces  205 ,  206 . The embodiment of FIG. 10 is similar to the embodiment of FIG.  3 . One of the primary distinctions is that the lapped panel  201  has a jagged, oblique edge  207  as indicated. The overlap portion  208  of the lapping panel  202  has a generally straight edge. Furthermore, the top surface  205  may be characterized as flat or generally smooth whereas the top surface  206  has an undulating surface. 
     Referring to FIG. 11, a top view of an alternative tile  10  is shown and comprises a lapped panel  211  and a lapping panel  212 . The lapped panel  211  and the lapping panel  212  have corresponding profiles  213 ,  214  and top surfaces  215 ,  216 . The embodiment of FIG. 11 is similar to the embodiment of FIG. 3 in overall function. One distinction is that the lapped panel  211  has a jagged, oblique edge  217  as indicated. The overlap portion  218  of the lapping panel  212  has a generally straight edge. The top surface  215  may be characterized as a roughened surface and the top surface  216  has an undulating surface. 
     Referring to FIG. 12, a top view of an alternative tile  10  is shown and comprises a lapped panel  221  and a lapping panel  222 . The lapped panel  221  and the lapping panel  222  have corresponding profiles  223 ,  224  and top surfaces  225 ,  226 . The embodiment of FIG. 12 is similar to the embodiment of FIG. 4 in overall function. One distinction is that the lapped panel  221  has a jagged, oblique edge  227  as indicated. The overlap portion  228  of the lapping panel  222  has a generally straight edge. The top surfaces  225 ,  226  may be characterized as having an undulating surface. 
     Referring to FIG. 13, a top view of an alternative tile  10  is shown and comprises a lapped panel  231  and a lapping panel  232 . The lapped panel  231  and the lapping panel  232  have corresponding profiles  233 ,  234  and top surfaces  235 ,  236 . The embodiment of FIG. 13 is similar to the embodiment of FIG. 4 in overall function. One distinction is that the lapped panel  231  has a jagged, oblique edge  237  as indicated. The overlap portion  238  of the lapping panel  232  has a generally straight edge. The top surface  235  has a generally smooth or flat surface and the top surface  236  has an undulating surface. 
     Referring to FIG. 14, a top view of an alternative tile  10  is shown and comprises a lapped panel  241  and a lapping panel  242 . The lapped panel  241  and the lapping panel  242  have corresponding profiles  243 ,  244  and top surfaces  245 ,  246 . The embodiment of FIG. 14 is similar to the embodiment of FIG. 3 in overall function. One distinction is that the overlap portion  248  of the lapping panel  242  has a jagged, oblique edge. A further distinction is that the top surface  245  is grooved and comprises a series of lands  240   a  and grooves  240   b  in an alternating fashion as illustrated. Each land  240   a  and groove  240   b  have corresponding widths  249   a,    249   b.  The top surface  226  has a smooth or generally flat surface. 
     Referring to FIG. 15, a top view of an alternative tile  10  is shown and comprises a lapped panel  251  and a lapping panel  252 . The lapped panel  251  and the lapping panel  252  have corresponding profiles  253 ,  254  and top surfaces  255 ,  256 . The embodiment of FIG. 15 is similar to the embodiment of FIG. 4 in overall function. One distinction is that the overlap portion  258  of the lapping panel  252  has a jagged or oblique edge. The top surface  255  has a grooved surface and the top surface  256  has a smooth or generally flat surface. 
     The present invention may be embodied in other specific forms without departing from its structures, methods, or other essential characteristics as broadly described herein and claimed hereinafter. The described embodiments are to be considered in all respects only as illustrative, and not restrictive. The scope of the invention is, therefore, indicated by the appended claims, rather than by the foregoing description. All changes which come within the meaning and range of equivalency of the claims are to be embraced within their scope.