Patent Publication Number: US-9840847-B2

Title: Curved staircase

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application claims priority to U.S. Patent Application Ser. No. 62/213,237 to Moritz O. Bergmeyer entitled “SPIRAL STAIRCASE”, filed Sep. 2, 2015, the disclosure of which is hereby incorporated entirely herein by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     Technical Field 
     This invention relates to a self-supporting curved staircase having a tetrahedral support structure. 
     State of the Art 
     Buildings all over the world incorporate spiral or curved staircases for their space savings and aesthetic appeal. However, spiral staircases often require a central column for support which makes them look like the scaffolding to build the stair was left in place. 
     While there are spiral or curved staircases that do not require a central column, these large double helix stairs are very expensive to build due to very heavy structural elements. Alternatively, they are often built using a curved wall for support. 
     A more natural light weight structure which imitates to some degree structures found in nature or crystals is a better solution. With the ability of bigger and faster computers, highly redundant structures as here proposed are able to be analyzed to meet building codes for strength and earthquake resistance that could not have been analyzed effectively even 40 years ago. 
     While there are many patents for circular staircases, most of them are similar to U.S. Pat. No. 3,667,176 issued to Donald R. H. Mackay, entitled Spiral Staircases and which discloses a spiral staircase having a rod through the center of the spiral in order to support the stairs. 
     Additional patents disclose self-supporting spiral or curved staircases, such as U.S. Pat. No. 6,112,480 issued to Scott A. Turner, entitled Modular Staircase. While this patent discloses, a self-supporting staircase, its design is very bulky and unattractive. 
     Accordingly, what is needed is a self-supporting curved staircase that is strong and aesthetically pleasing. 
     DISCLOSURE OF THE INVENTION 
     The disclosed invention relates to a self-supporting curved staircase having a tetrahedral support structure. 
     An embodiment of self-supporting curved staircase may include a plurality of stair segments, wherein each of the stair segments includes a plurality of rods coupled to a plurality of connecting nodes. The plurality of rods are arranged in a skewed tetrahedral geometry. The skewed tetrahedral geometry causes the plurality of stair segments to form a curved structure when the plurality of stair segments are coupled together. The plurality of rods form a spine on an underside of the plurality of stair segments. Treads are coupled to each of the stair segments. 
     An additional embodiment of self-supporting curved staircase may include a support structure, wherein the support structure comprises a plurality of segments having at least 9 rods coupled to at least 3 connecting nodes. The at least 9 rods and the at least 3 connecting nodes form a tetrahedral geometry. The tetrahedral geometry causes each of the plurality of segments to have a non-symmetrical shape. When the plurality of segments are coupled together the non-symmetrical shape causes the plurality of segments to form a helical support structure. A plurality of treads are coupled to the support structure. 
     A further embodiment of a self-supporting curved staircase may include a support structure comprising a plurality of linear support locations coupled at a plurality of connecting nodes. The plurality of linear support locations and the plurality of connecting nodes form a plurality of tetrahedrons. The plurality of linear support locations and the plurality of connecting nodes also form a spine on an underside of the self-supporting curved staircase. A plurality of treads are coupled to the support structure. 
     The foregoing and other features and advantages of the invention will be apparent to those of ordinary skill in the art from the following more particular description of the invention and the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In order that the invention may be clearly understood, it will now be described by way of example with reference to the accompanying drawings, wherein 
         FIG. 1  is an isometric view of a first embodiment of a self-supporting curved staircase structure. 
         FIG. 2  is a side view of a first embodiment of a self-supporting curved staircase structure. 
         FIG. 3  is a bottom view of a first embodiment of a self-supporting curved staircase structure. 
         FIG. 4  is a partially exploded view of a stair segment of a first embodiment of a self-supporting curved staircase structure. 
         FIG. 5A  is a vertical cross section taken along a center of a self-supporting curved staircase coupled to a floor. 
         FIG. 5B  is a vertical cross section taken along a center of a self-supporting curved staircase coupled to a landing. 
         FIG. 6  is a top view of a self-supporting curved staircase coupled to a floor. 
         FIG. 7A  is a top view of a first mounting plate for mounting a self-supporting curved staircase to a landing. 
         FIG. 7B  is a top view of a second mounting plate for mounting a self-supporting curved staircase to a landing. 
         FIG. 8  is an isometric view of a stair segment of a second embodiment of a self-supporting curved staircase. 
         FIG. 9  is an isometric view of a third embodiment of a self-supporting curved staircase. 
         FIG. 10  is a front view of a third embodiment of a self-supporting curved staircase. 
         FIG. 11  is a bottom view of a third embodiment of a self-supporting curved staircase. 
         FIG. 12  is an isometric view of an embodiment of a self-supporting curved staircase structure with a railing. 
         FIG. 13  is a top view of an embodiment of a self-supporting curved staircase structure with a railing. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     As discussed above, embodiments of the present invention relate to a self-supporting curved staircase including a curved form such as a winder form, helical or double helix form with no center support column or the like. The curved staircase is supported by a three dimensional curved space frame having a tetrahedral geometric structure. 
     Tetrahedral structures are described herein for support of a curved staircase. Sweeping larger diameter stairs may use a modified tetrahedron structure for support. The curved staircase structure as described herein may be analyzed for structural and earthquake stability and strength requirements. Further, the described curved staircase is efficient, lightweight and aesthetically pleasing. 
     For the purposes of this application, the following definitions will apply. A tetrahedron is a 3 dimensional object having four triangular faces; a triangular pyramid. Tetrahedral means pertaining to or having the form of a tetrahedron. Helical means having the shape or form of a helix. Double helix means a pair of parallel helices intertwined about a common axis. 
       FIGS. 1-3  illustrate an embodiment of a self-supporting curved staircase structure  100  for use with large outside diameter curved stairs, particularly an outside diameter in the range of about 5 feet to about 30 feet or more. With larger outside diameter curved stairs, there is a need for more structural stability and less vibration of the support structure. 
     The self-supporting curved staircase structure  100  is formed from multiple stair segments  200  coupled together so as to form a helical staircase or double helix shape. The stair segments  200  are a stair step section that holds a single tread. The stair segments  200  are coupled front to back in order to form a full staircase. The front of the next stair segment  200  is coupled to the back of the previous stair segment  200 . 
     Each stair segment  200  is formed from multiple support rods  114 . Support rods  114  are illustrated as thin rods formed from metal or other strong, durable material. Support rods  114  may be any shape or size desired. They may be formed as thin rods as illustrated or they may be formed as flat strips of metal, hollow tubes, wooden dowels, polycarbonate rods, creases in sheet metal or the like. Provided that the support rods  114 , are strong enough to support the curved staircase  100 . 
     Each support rod  114  may be formed as a single piece or may be multiple pieces coupled together via welding, adhesive, male/female connector or the like. It is anticipated, however, that forming the support rods  114  as a single piece will provide the most strength to the curved staircase  100 . 
     In the embodiment of the curved staircase  100  illustrated in  FIGS. 1-3 , there are 18 support rods  114  for each stair segment  200 . In alternate embodiments of the curved staircase  100 , there may be more or fewer support rods  114 . 
     The support rods  114  are connected together at nodes  118 . Nodes  118  are connection locations for multiple support rods  114 . The nodes  118  may be formed by welding multiple support rods  114  together, or else the nodes  118  may have threaded, clip, compression or other connections in order to connect the support rods  114  together. 
     In the embodiment of the curved staircase  100 , there are illustrated  5  nodes  118  for each stair segment  200 . 
     The support rods  114  and the nodes  118  are arranged so as to create a tetrahedral geometry or in other words, they are arranged so as to form multiple tetrahedrons. The tetrahedrons utilized in the self-supporting curved staircase are skewed or modified in order to cause the staircase to form a helical arrangement or a double helix. The skewed tetrahedrons cause the stair segments  200  to have a non-symmetrical geometry which, when multiple stair segments  200  are coupled adjacent one another, causes the staircase to turn and form a double helix. Additionally, the support rods  114  and the nodes  118  are arranged so as to create a spine  120  which runs along the middle underside of the curved staircase structure  100  adding additional strength. The spine  120  is a line of support rods  114  with node  118  intersections that are arranged so as to create a line that follows the curve of the curved staircase  100 . 
     Each stair segment  200  has a tread  116  placed on top of the top of the stair segment  200 . The treads  116  are where the user will step as he/she ascends or descends the curved staircase  100 . The treads  116  may be formed from any material desirable, including wood, marble, stone, metal, glass or the like. The treads  116  may be formed in any size or shape desirable. They may be formed as squares, rectangles, ovals, rounded rectangles or the like. However, the treads  116  must fit in the space provided on the stair segment  200 . Additionally, the treads  116  should be strong enough to support the weight of a user and lightweight enough to not add significant stress to the staircase support  100 . 
     Treads  116  are coupled to the underlying staircase support structure  100 . Treads  116  may be coupled to the support structure  100  using bolts, screws, nails, adhesive, welding, or the like. The treads  116  may appear to be floating on the structure or fit tight to the corners of the structure. 
     The inner and outer edges of the curved staircase structure  100  each have a cheek which rises above the level of the tread  116 . The inner cheek  112  is on the inside of the curved staircase  100  curve. The outer cheek  110  is on the outside of the curved staircase  100  curve. The inner cheek  112  and the outer cheek  110  are formed from multiple support rods  114  coupled with nodes  118 . The cheeks  112  and  110  provide additional strength and support to the curved staircase  100 . Additionally, they provide a location for balusters or the like to be coupled to the curved staircase  100 . 
       FIGS. 1-3  also illustrate a bottom support plate  122 . The bottom support plate  122  is a piece of sheet metal or other strong material which is coupled to the lower end of the curved staircase  100 . The bottom support plate  122  is also coupled to the floor of the building the curved staircase  100  is installed in. The bottom support plate  122  may be formed in any size or shape desired. 
     The top of the curved staircase  100  would be coupled to an upper landing using an upper support plate illustrated in  FIGS. 5B, 7A and 7B . 
       FIG. 2  also illustrates that the curved staircase  100  widens at a lower or bottom portion  124  of the staircase  100 . Additionally, the curved staircase  100  widens parallel to the floor at an upper, top or landing portion  126  of the curved staircase  100 . In other words, the inner cheek  112  and the outer cheek  110  are farther apart on the bottom few steps and the top few steps. The wider portions at the top  126  and bottom  124  portions of the staircase  100  provide added strength and stability to the structure. 
     The structure supporting the curved staircase  100  also deepens at the top portion  126  and bottom portion  124  of the staircase  100 . In these portions of the staircase  100 , the spine  120  is farther from the treads  116  than in the other segments of the staircase  100 . This deepening of the support structure of the staircase  100  also adds strength and stability to the staircase  100 . 
       FIG. 4  illustrates an embodiment of a stair segment  200  of a first embodiment of a self-supporting curved staircase. Each segment  200  is composed of 18 rods and 5 nodes. Each segment is able to be fabricated and will fit against a lower tread segment for final welding or attachment to the lower segment as well as welding or otherwise coupling of sufficient balusters and handrail to meet building safety codes. Treads will also be coupled to the stair segment  200 . 
     Stair segment  200  includes 18 rods labeled  225 ,  226 ,  227 ,  228 ,  229 ,  211 ,  212 ,  213 ,  214 ,  216 ,  217 ,  218 ,  219 ,  221 ,  223 ,  224 ,  222 , and  220 . Rod  232  is a support rod from the back of the previous stair segment  200 . The  5  nodes in this figure are labeled  234 ,  235 ,  233 ,  236  and  237 . 
     The segment of the stair is composed of two cheek structures with 4 rods each (the inner cheek  112  composed of rods  211 ,  212 ,  213 , and  214 ; and the outer cheek  110  composed of rods  216 ,  217 ,  218 , and  219 ). As discussed previously, the cheeks are coupled to the edges of the stair segments  200  on the inside and the outside. Rods  211  and  216  are coupled from the front of the stair segment  200  illustrated to the front of the stair segment  200  above. This causes the cheeks  110  and  112  to create side edges on the treads. 
     The spine section  238  is coupled to the bottom of the stair segment  200 . The spine section is composed of rods  220 ,  221 ,  222 ,  223 , and  224 . Rod  222  couples to similar rods on other segment sections to form the spine  120  (see  FIGS. 1-3 ). The spine section  238  in  FIG. 4  comprises 5 rods. Rod  222  becomes the spine, while rods  220 ,  221 ,  223  and  224  extend at an angle downward from the stair segment  200  to support rod  222 . 
     The remaining pieces of the segment  200  are connecting rods  225 ,  226 ,  227 ,  228 , and  229 . For orientation rods  228 ,  229 ,  214 ,  219 ,  223 , and  224  are in the same vertical plane (as noted by the vertical extent of the plane shown as  230 ). Rods  225 ,  226 ,  228 ,  217 , and  212  are in the horizontal plane shown as  231  under the tread itself. Rod  232  from the lower segment which is the alignment and connecting rod to the newer segment. Planes  230  and  231  are included for illustrative purposes only and are not actual physical surfaces. 
     Each segment has 5 nodes  233 ,  234 ,  235 ,  236 , and  237  which are the  5  corners of the vertical plane of the front of the segment. Nodes  233 ,  234 ,  235 , and  236  are determined by the geometry of the stair dimensions. Node  237  is a variable height from rod  232  which is determined by structural analysis for strength and rigidity of the structure in large diameter curved stairs. 
     As illustrated in the figures, the tetrahedral structure in the disclosed staircase is not formed from equilateral tetrahedrons, instead the tetrahedral structure is a skewed tetrahedral structure wherein the lengths of the legs of the tetrahedrons or the proportions of some of the lengths of the legs of the tetrahedrons to the remaining legs of the tetrahedrons have been altered in order to form a curved or helix. As the proportions of the lengths of the legs of the tetrahedrons change, the shape and curvature of the curved staircase are altered. 
       FIGS. 5A and 5B  are vertical sections through the stair near the center of the stair showing the main stair elements, the spine, and how the stairs are attached to the floor as shown in  FIG. 5A  and how the stair is attached to the upper building structure as shown in  FIG. 5B .  FIG. 5A  shows the vertical section through the bottom  5  tread segments of the stair.  FIG. 5A  shows bottom metal plate  122 , attachment points  316  to floor structure  319 , and typical treads  116 . Rods  114  act as spine elements. The last  3  stair segments have nodes  118  which touch floor structure  319 . The bottom two steps could be splayed outward to add additional strength to the stair. The spine structure may contact the bottom metal plate  122  for three or more stair segments. 
       FIG. 5A  illustrates a bottom portion of a self-supporting curved staircase  100 . The bottom portion of the curved staircase  100  is coupled to floor  319 . Floor  319  is the surface to which the lowest end of the curved staircase  100  is coupled. Curved staircase  100  is coupled to floor  319  by bottom support plate  122 . 
     Bottom support plate  122  is a metal plate to which the support structure of the curved staircase  100  is coupled. The support structure  100  may be coupled to the bottom support plate  122  through bolts, screws, welding, adhesive, epoxy or the like. Provided that the connection is strong enough to withstand the forces applied to the curved staircase  100  during use. 
     The bottom support plate  122  is coupled to floor  319  through couplers  316 . Couplers  316  may be screws, bolts, clips, nail, adhesive or the like. Couplers  316  may be any device or substance that can securely attach the bottom support plate  122  to the floor  319  surface. Provided that the connection is strong enough to withstand the forces applied to the curved staircase  100  during use. Additionally, as many or as few couplers  316  may be used to couple to the bottom support plate  122  to the floor  319  as is desired or necessary to secure the bottom support plate  122  to the floor  319 . 
       FIG. 6  illustrates and embodiment of a bottom support plate  400 . The metal plate  414  is shaped with a wide end  416  which supports the lowest stair of the curved staircase. The metal plate  414  narrows towards the other end which is used to support the next few stair segments. The wide end  416  of the bottom support plate  400  may be shaped as an elongated diamond or else the bottom support plate  400  may be formed in any shape desired, provided that the bottom support plate  400  is strong enough and large enough to support the curved staircase and that the bottom support plate  400  does impede with the architecture of the building if at all possible. 
       FIG. 6  also illustrates rods from the curved staircase coupled to the bottom support plate  400  at nodes  402 ,  404 ,  406 ,  408 ,  410 , and  412 . The rods may be welded, bolted, or coupled to the bottom support plate  400  with adhesive or the like. Additionally, the rods may be screwed into threaded female receivers formed in the bottom support plate  400 . 
     While the bottom support plate  400  illustrated in  FIG. 6  is one possible shape of a bottom support plate  400 ,  FIG. 3  illustrates an alternatively shaped bottom support plate  122 . Additionally, the bottom support plate may be shaped in anyway desired, so long as it serves its purpose. The bottom support plate may also be formed from any material desired, provided it can be coupled to the curved staircase  100  and has enough strength to couple the curved staircase  100  to the floor. 
       FIG. 5B  shows a vertical section through the top  4  stair treads  116  and second floor landing  320 . At the top of the stair  100  where the stair  100  is attached to the building, nodes  118  for the top two or more stair segments would be deepened and/or split to form a triangular shape to add additional strength and rigidity to the stair (rods  114  are the typical spine rods of the stair structure). In addition the top two stair segments could be splayed to add to the strength and rigidity of the stair. Attachment of the top of the stair would have the top segment rods welded or otherwise coupled to a metal plate  315  so as to distribute the stresses of the stair onto the supporting building structure  320 . 
       FIGS. 7A and 7B  show two different upper support plates  500  which are used to mount the top of the staircase to the upper landing  320  (see  FIG. 5B ).  FIG. 7A  illustrates an upper support plate  500  which splays out the upper  1  or  2  stair segments making them wider.  FIG. 7B  illustrates an upper support plate  500  that supports the stairs by lowering node  118  (see  FIG. 5B ) and coupling it to the side of the upper landing. 
       FIG. 7A  illustrates an upper support plate  500  formed in the shape of a triangle with a flat top. With additional reference to  FIG. 5B , the upper support plate  500  is a flat metal plate that is mounted to the upper landing  320  and to which the staircase is coupled. In the embodiment illustrated in  FIG. 7A , node  118 , which is the node at the bottom or along the spine of the staircase is split or bifurcated in order to provide additional support to the staircase. The bifurcated node  118  is coupled to the upper support plate  500  at locations  520  and  522 . The inner cheek and outer cheek of the staircase are coupled to the upper support plate  500  at locations  524 ,  526 ,  528  and  530 . Thereby, securing the staircase to the upper support plate  500  which is secured to the upper landing  320 . Outline  540  illustrates the shape of a normal stair before the stair is flared out in order to couple to locations  520 ,  522 ,  524 ,  526 ,  528  and  530 . The flared stair increases structural strength of the staircase and is aesthetically pleasing. 
       FIG. 7B  illustrates an upper support plate  500  formed in the shape of an upside down triangle. With additional reference to  FIG. 5B , the upper support plate  500  is a flat metal plate that is mounted to the upper landing  320  and to which the staircase is coupled. In the embodiment illustrated in  FIG. 7B , node  118 , which is the node along the spine of the staircase, is coupled to the upper support plate  500  at location  520 . The inner cheek and outer cheek are coupled to the upper support plate  500  at locations  524 ,  526 ,  528  and  530 . Outline  540  illustrates the shape of a normal stair before the stair is flared out in order to couple to locations  520 ,  524 ,  526 ,  528  and  530 . The flared stair increases structural strength of the staircase and is aesthetically pleasing. 
     The staircase may be coupled to the upper support plate  500  by screws, bolts, welding, epoxy, adhesive or the like. 
     The upper support plate  500  may be formed in any size or shape desired, i.e., circular, oval, rectangular, square and the like. Additionally, any artistic embellishment desired may be added, so long as it does not impede the strength and purpose of the upper support plate  500 . 
     The upper support plate  500  may also be formed from any material desired, so long as the upper support plate  500  is strong enough to support the weight of the curved staircase when it is in use. It is likely that a thin metal plate formed in various shapes will be most often used as an upper support plate  500 . 
       FIG. 8  shows an isometric view of an alternate embodiment of a stair segment  600 . Stair segment  600  has 9 rods per tread  611 ,  612 ,  613 ,  614 ,  615 ,  617 ,  618 ,  619 , and  620  with 3 connecting nodes  624 ,  626 , and  625  arranged in a tetrahedral structure for curved stairs that are of a smaller diameter, particularly an outside diameter in the range of from about 4 feet to about 10 feet. 
     Node  625  is the spine node and it, in combination with rod  613 , are coupled to the spine node and rod from other steps to form the spine along the underside of the self-supporting curved staircase. 
     The stair segment  600  comprises an outer cheek  610  with 4 rods  611 ,  612 ,  613 , and  614 , arranged in a sloped manner. Rod  611  extends from the front of stair segment  600  to the front of the next stair segment above. Rods  612 ,  613 , and  614  are coupled in a triangular shape. 
     Inner cheek  616  is comprised of only rod  615 . Rod  615  of inner cheek  616  is sloped to go from the front of the lower step to the rear of the step it is a part of. 
     There are 4 lateral rods connecting inner cheek  616  and outer cheek  610 . The 4 lateral connecting rods are  617 ,  618 ,  619 , and  620 . Additionally, rod  620  inter connects two stair segments. Plane  622  shows the location of the tread which is supported by rods  617 ,  618 , and  612 , and plane  623  shows the surface of the vertical step element structure formed by rods  619 ,  618  and  614 . Planes  622  and  623  are included for illustrative purposes only and are not actual physical surfaces. 
       FIGS. 9-11  illustrate an alternate embodiment of a self-supporting curved staircase  700  formed from sheet metal and internal stiffening rods, similar to the rods in previous embodiments. As in previous embodiments, this embodiment includes a plurality of linear support locations  714 . In previous embodiments, these linear supports were rods. In this configuration, however, the rods of the underbody structure are replaced with creases in the sheet metal used to form the curved staircase  700 . Both the  18  rod and 9 rod configurations may be used with the sheet metal curved staircase  700 . 
     The linear support locations  714  may be slight creases in the sheet metal, where the angle formed by the crease is obtuse. Alternately, the linear support locations  714  may be formed as acute angles in the sheet metal, depending on the desired configuration. 
     Additionally, a spine  712  is also formed in the sheet metal in order to increase the strength of the curved staircase  700 . 
     An outer cheek  718  and an inner cheek  720  are also formed by bending the sheet metal parallel to the tread location of the curved staircase  700 . 
     Treads  710  are placed on top of the sheet metal structure with the treads  710  overlapping and hiding the outer cheek  718  and inner cheek  720 . The treads  710  may be formed in any shape or size desired. Additionally, the treads  710  may be formed from any material desired. 
     The treads  710  may also, as illustrated, be formed from two pieces which are coupled parallel one another on the sheet metal structure. 
     The treads  710  may be flat and sit on top of the inner cheek  720  and the outer cheek  718 , or the treads  710  may have a lip which overlaps the cheeks. 
     The treads  710  may be coupled to the sheet metal structure through screws, bolts, adhesives or the like. 
     The creases or linear support locations  714  meet at nodes  716  similarly to the rods in previous embodiments. Nodes  716  are indentations in the sheet metal where multiple linear support locations  714  end. 
     Additionally, stiffening rods  722  may be placed inside the sheet metal structure to add strength and stiffness to the curved staircase  700 . The stiffening rods  722  are coupled between the inner cheek  718  and the outer cheek  720  of the sheet metal staircase. The stiffening rods  722  may be metal rods, metal tubes, wooden dowels, panels or the like. 
     The sheet metal self-supporting curved staircase  700  may also widen at a top and bottom portion of the staircase  700  in order to add additional strength and stability to the staircase  700 . 
       FIGS. 12-13  illustrate an additional embodiment of a self-supporting curved staircase  800 . The curved staircase  800  includes rods  814  and nodes  818  that are similar to and provide similar function as previously discussed with regard to rods  114  and nodes  118 . Additionally, the staircase  800  includes a spine  820  formed by the rods  814  and nodes  818 . The curved staircase  800  also includes a bottom support plate  822  identical to those previously discussed. An upper support plate would be used to couple the self-supporting curved staircase  800  to an upper landing as described previously. 
     In this embodiment, however, the inner cheek  812  and outer cheek  810  have balusters  824  coupled along them. These balusters  824  may be formed in any shape desired. The balusters  824  illustrated are curved metal strips which bow out at the bottom and taper towards the center of the staircase at the top. The balusters  824  provide added strength and stability to the overall curved staircase structure  800 . 
     The balusters  824  have a railing  826  coupled along a location near the top of the balusters. The railing  826  may be formed from metal, wood, glass or the like. The railing  826  may have a cross-section that is circular, square, rectangular or may have a decorative shape. The railing  826  also increases the strength of the staircase  800 . 
     The railing  826  may be coupled to the balusters  824  by welding, screws, bolts, adhesives, epoxies or the like. Provided the method of coupling the railing  826  to the balusters  824  is sufficiently strong to prevent the railing  826  from coming loose. 
       FIG. 13  is a top view of a curved staircase  800  with balusters  824  and a railing  826 . As illustrated in this figure, the curved staircase  800  may widen at a top and bottom location in order to increase the strength and stability of the staircase  800 . 
     In alternate embodiments of a self-supporting curved staircase with balusters and a railing, artwork may be applied to the treads and/or balusters. Portions of the artwork may be applied to each tread or each baluster in order to allow the user to see the entire image from a distance. 
     The structure described in this application with respect to curved staircases may also be applied to other curved structures such as curved ramps, curved rooftops and the like and the description herein should not be limited to staircases. 
     The curved staircases described above may also be formed in many different ways. The curved staircase may be welded, bolted, epoxied or the like. Additionally, the curved staircase may be formed with a 3d printer, wherein the  3   d  printer would form the staircase from metal, polymers or other materials which are strong enough to meet the structural demands of the staircase. 
     The curved staircases disclosed herein may be formed from any material desirable. Examples of materials include wood, glass, metal, polymers, plastics, carbon fiber, fiberglass, composites and the like. Additionally, the staircases described herein may be formed from multiple types of materials, i.e. the stair tread may be formed from glass while the rods may be formed from metal or carbon fiber and the upper support plate and floor plate may be formed from wood or metal. 
     Additional embodiments of the curved staircases disclosed above may be formed with from fewer or greater numbers of rods and nodes, so long as those rods form tetrahedral structures such as those discussed above. Therefore, though the figures disclose a certain number of rods and nodes, the figures were included for exemplary purposes only and are not meant to be limiting in anyway. 
     Alternate embodiments may also include the stair case being formed from sheet goods, i.e. sheet metal, plastic sheeting, or the like. The sheet goods would be used to form the tetrahedral planes of the structure. Solid materials, i.e. foam, plastic, concrete, foam coated with fiberglass or the like, may also be used to form the stair case. Solid materials would be used to follow the shape of the described stair in either a single form or post tensioned stair segments. 
     The embodiments and examples set forth herein were presented in order to best explain the present invention and its practical application and to thereby enable those of ordinary skill in the art to make and use the invention. However, those of ordinary skill in the art will recognize that the foregoing description and examples have been presented for the purposes of illustration and example only. The description as set forth is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the teachings above.