US3841039A

US3841039A - Modular vertex structure

Info

Publication number: US3841039A
Application number: US00001583A
Authority: US
Inventors: P Farnsworth
Original assignee: Individual
Current assignee: Individual
Priority date: 1970-01-09
Filing date: 1970-01-09
Publication date: 1974-10-15
Anticipated expiration: 1991-10-15

Abstract

1. A POLYHEDRAL STRUCTURE HAVING A PLURALITY OF VERTICES AND A PLURALITY OF EDGES MEETING AT EACH OF THE VERTICES AND DEFINING THE FACES OF THE POLYHEDRAL STRUCTURE COMPRISING: A PLURALITY OF SUPPORT FRAME MEMBERS DEFINING THE EDGES WHICH DEFINE AT LEAST CERTAIN FACES OF THE POLYHEDRAL STRUCTURE, EACH OF SAID SUPPORT FRAME MEMBERS HAVING AT LEAST ONE LEG MEMBER OF A PAIR OF LEG MEMBERS EXTENDING AT LEAST PART WAY ALONG EDGES DEFINING SUCH FACES AND AT LEAST ONE CURVED PORTION DEFINING AN ARC OF A SUBSTANTIALLY CIRCULAR AREA CENTERED IN THE FACE OF THE POLYHEDRAL STRUCTURE DEFINED BY SUCH EDGES, THE PLURALITY OF SUPPORT FRAME MEMBERS AT EACH FACE JOINED TOGETHER WITH THE CURVED PORTIONS THEREOF COMPLETELY SURROUNDING SAID SUBSTANTIALLY CIRCULAR AREA THEREOF, AT LEAST A PAIR OF SUBSTANTIALLY CIRCULAR MEMBRANE WALL MEMBERS CLOSING SAID AREA WITHIN SAID SURROUNDING SUPPORT FRAME MEMBERS AT EACH OF SUCH FACES, MEANS FOR COAXIALLY MOUNTING EACH PAIR OF SAID MEMBRANE WALL MEMBERS FROM SAID CURVED PORTIONS OF SAID SURROUNDING SUPPORT FRAME MEMBERS COMPLETELY AROUND THE PERIPHERY OF SAID AREA AND SEALING THE SPACE BETWEEN EACH PAIR OF SAID MEMBRANE WALL MEMBERS, AND MEANS FOR ADJUSTING THE PRESSURE WITHIN THE SPACE BETWEEN EACH PAIR OF SAID MEMBRANE WALL MEMBERS CAUSING SAID MEMBRANE WALL MEMBERS TO TENSION SAID SURROUNDING SUPPORT FRAME MEMBERS RADIALLY INWARD OF SAID AREA.

D R A W I N G

Description

Farnsworth, 111

[45] Oct. 15, 1974 I 1 MODULAR VERTEX STRUCTURE [76] Inventor: Philo T. Farnsworth, III, PC. Box

251, Bolinas, Calif. 94924 [22] Filed: Jan. 9, 1970 [21] Appl. No.: 1,583

[52] US. Cl 52/81, 52/DIG. 10, 52/73, 52/222 [51] Int. Cl E041) 5/43 [58] Field of Search 52/79, 80,81, 73, 121, 52/DIG. 10; 292/296.61

[56] References Cited UNITED STATES PATENTS 3,633,325 l/-l972 Bartoli 52/73 1,073,859 9/1913 Laskey 292/256.6l 3,127,699 4/1964 Wasserman 52/81 3,153,235 10/1964 Chatelain 52/2 3,192,575 7/1965 Rosenan 52/171 FOREIGN PATENTS OR APPLICATIONS 741,047 1966 Canada 52/81 1,088,671 France 52/81 1,568,630 1969 France 52/73 45,677 1966 Germany 52/81 843,529 1960 Great Britain 52/80 1,009,850 1965 Great Britain 52/81 OTHER PUBLICATIONS Philosophical Transactions of the Royal Society of London, Series A; Volume 246, p. 440, 441, 443 & 445, from Uniform Polyhedra by Coxeter. Domebook 2, 1971, pages 9 and 10.

Primary Examiner-Frank L. Abbott Assistant Examiner-H. E. Raduazo Attorney, Agent, or Firm -Limbach, Limbach & Sutton 5 7 ABSTRACT A polyhedral building structure is disclosed with formed side walls and modular at the vertices thereof. The side walls can be in the form of frustoconical sections and the modular vertices can be in the form of structural members including first and second leg members lying along edges that meet in a vertex and an arcuate portion extending between spaced apart portions of the leg members. The leg members can extend partially or half-way to the next adjacent vertex. Frustoconical sections can be located at and serve as the vertex portion of the polyhedral structure. A building with a conical base portion projecting through one of the frustoconical sections can be utilized and modular pie-shaped floor sections positioned within the building structure over the pedestal. The frustoconical sections can include one or more evacuated or pressurized chambers formed by circular membranes therein which provide integrity to the structure.

10 Claims, 15 Drawing Figures PATENIEBUBT 1 512m SHEEI 1 0f 6 IINVENTOR. PHILO T. FARNSWORTH H1 ATTORNEYS PATENTEB MT 1 51974 SHEEY BM 6 FIGB INVENTOR, PHILO T. FARNSWORTH I11 Pmmmm 3.841.039

SHEET 3 OF 6 FIG.5

INVENTOR.

PHILO T. FARNSWORTHJJI ATTORNEYS PAIENIEBWI l SIS SHEEI BF 6 FIG.7

INVENTOR. BYPHILU T.FARNSWORTH11I ATTORNHEYS SHEEI 5 OF 6 INVENTOR. PHILO IFARNSWORTHIII ATTORNEYS PAIENIEB w 3.841 .039

SHEET 6 0F 6 INVENTOR.

PHILO T. FARNSWORTH 111 ATTORNEYS MODULAR VERTEX STRUCTURE BACKGROUND AND DESCRIPTION OF INVENTION I The present invention is directed to a polyhedral building structure, and. more particularly, to a modular structure for low cost housing.

The need for low cost housing is an ever present need throughout the world. It has long been recognized that the best way to provide low cost housing is to produce a functional structure that can easily be assembled on the site from a minimum number of different components which are mass produced at a factory. To date, attempts to achieve this result have not been successful.

Broadly stated, the present invention, to be described in greater detail below, is directed to a polyhedral building structure with preformed side walls corresponding to the faces of a polyhedron and modular at the intersections of the side walls corresponding to the vertices of the polyhedron.

In accordance with one aspect of the present invention, the side walls, as well as certain of the vertices as described below, are in the-form of frustoconical sections formed of a plurality of subassemblies.

In accordance with another aspect of the present invention, the subassemblies join together to form the vertex of a member whereby a structure is provided which is modular at the vertex. These modular subassemblies include first and second leg members lying along edges that meet in a vertex such that assembly of the appropriate number of said subassemblies at a point will establish the vertex. With the leg members equal in length to half the distance between vertices of a polyhedral structure, the entire structure can be formed by assembling each of the vertices fromthe appropriate number of said subassemblies and then connecting the adjacent vertices together to complete the polyhedral construction.

In accordance with still another aspect of the present invention, the subassemblies for forming modular vertices include arcuate portions extending between spaced apart portions of the leg members whereby the arcuate portions form truncated conical sections when a plurality of said subassemblies are joined together.

In accordance with still another aspect of the present invention, a structure is formed in accordance with the aspect of the preceding paragraph and wherein membranes, preferably circular, are supported in the truncated conical sections to serve as wall members for the polyhedral structure and at the same time act as structural support for the individual truncated conical sections.

Where membranes are provided in the truncated conical'sections. a plurality of said membranes can be supported there and the pressure of the space between the membranes controlled with respect to atmospheric pressure to provide forces holding the modular members together and thus contribute to the structural integrity of the polyhedral configuration. With membranes supported together at their periphery, pressurization of the space between the membranes establishes forces pulling the periphery of the membranes inwardly to support the truncated conical section from which the membranes aremounted. With the peripheries of a pair of membranes spaced apart along the surface of a truncated conical section, the

space between the membranes can be partially evacuhedral structure.

In accordance with still another aspect of the present invention, frustoconical sections can be located at and serve as the vertex portions of polyhedral structures, and where this aspect of the present invention is utilized in conjunction with frustoconical sections forming the faces of the corresponding polyhedral structure, a building construction is provided with frustoconical sections of two different diameters.

Numerous construction features and advantages flow from use of modular members in accordance with the aspects of the invention described above. Accordingly, a building with a conical base portion projecting through one of the frustoconical sections can be utilized and a modular pie-shaped floor section positioned within the building over the pedestal. A building of this nature can be manufactured in modular form in a factory utilizing mass production techniques and the buildingquickly and easily assembled on the site with little or no preparation of the building site or variation in the terrain of the building site. I These and other features and advantages will become more apparent upon a perusal of thefollowing specification taken in conjunction with the accompanying drawings wherein similar characters of reference refer to similar structures in each of the several views.

IN THE DRAWINGS: A f

FIG. I is a perspective view of a conic form of tetrahedron partially broken away and showing modular formation of the structure. at the vertices.

FIG. 2 is a view similar to FIG. I but of a conic form of hexahedrom FIG. 3 is a view similar to FIG. 1 but of a conic form of octahedron.

FIG. 4 is a view similar to FIG. 1 but of a conic form of dodecahedron.

FIG. 5 is a view similar to FIG. 1 but of a conic form of icosohedron and partially broken away showing a building constructed in accordance with the present invention.

FIG. 6 is a perspective view, partially exploded, illustrating structure for formation of the structure illus-' trated in FIG. 5.

FIG. 7 is a perspective view of an alternativeembodiment of a portion of the structure :shown in FIG. 6.

FIG. 8 is an enlarged elevational sectional view of a portion'of the structure shown in FIG. 5.

FIG. 9 is an enlarged section view of a portion of the structure shown in FIG. 8 delineated by line 99.

FIGS. 10 and 11 are viewssimila'r to FIG. 8 showing other embodiments of this invention.

FIG. l2is a schematic view illustrating one form of modularity of the present invention.

FIGS. 13 and 14 are views similar to FIG. 12 showing other modularity constructions of the present invention.

FIG. 15 is a perspective view of still another embodiment of the present invention.

It will be appreciated from the foregoing and the detailed description hereinafter that the present invention can be utilized in a great many applications. However, it is ideally suited as a building structure and certain aspects particularly for low cost housing and therefore will be described for illustrative purposes with respect thereto.

Referring now to the drawings with particular reference to FIGS. 1-5, there are shown polyhedral building structures of different configurations embodying the present invention.

With particular reference to FIG. 1, there is shown a building structure in a conic form of tetrahedron which geometrically has four principal sides which meet along edges that intersectin four vertices. The conic tetrahedral structure 10 has four frustoconical sections 11 which form the faces of the tetrehedron and which, as shown, are built up from structural frame members 12 in modular form. Each of the structural members 12 includes first and

second leg members

13 and 14 which are joined together at a vertex 16 of the tetrahedral structure 10 and extend along edges half way to an adjacent vertex 16 whereby three structural members, one at each of the three vertices associated with each frustoconical section, form the edges of the tetrahedron associated with each frustoconical section. In the structural member 12, an arcuate portion 15 defining one portion of the conical surface of the frustoconical section 11 extends between

leg members

13 and 14. Each vertex 16 of the conic tetrehedral structure is thereby made up of three structural members 12.

The wall portions 17 of the building structure can be formed in a variety of ways including disc members or membranes supported within the frustoconical sections as described in greater detail below with reference to FIG. 8. As generally illustrated in FIG. 1, the wall portions can comprise a pair of circular or disc members 18 which are fixedly secured at their periphery to the arcuate portions of the structural members 12 making up the frustoconical sections 11 and provided with means, such as inflating valves, for pressurizing the space between members 18. With pressure applied, the disc members pull radially inwardly on the frustoconical sections adding additional support strength to the building structure. By selective inflation and selection of materials for the discs 18, the wall portions 17 of the frustoconical sections can selectively pass light or sound and actually act as a lens structure.

The tetrahedral building can be assembled in any one of a number of different ways depending upon the materials selected for the structural members 12. For example, depending upon their composition the structural members 12 can be bolted,'welded. or glued together to form vertices l6 and the free end of the leg members extending therefrom connected to leg members from adjacent vertices by any convenient manner.

' Typically the strength of the building structure which structures described hereafter frustoconical sections making up a portion of the polyhedra can be utilized.

of octahedron 24 having eight frustoconical sections 25 (FIG. 3), a conic form of dodecahedron 28 having twelve frustoconical sections 29 (FIG. 4), and the conic form of icosohedron 30 having twenty frustocon ical facial sections 31.

The conic form polyhedral structures illustrated in FIGS. 1-4 can include frustoconical minoror vertex sections at the vertices of the polyhedron tangent to the frustoconical major or facial sections described. For purposes of illustration, the conic form of icosohedron illustrated in FIG. 5 includes such frustoconical minor sections. As illustrated there, the conic form of icosohedron includes twenty major frustoconical facial sections 31 corresponding to the twenty faces of an icosohedron and twleve minor frustoconical vertex sec tions 32 of a diameter different from the frustoconical facial sections 31 and located at the vertices of the corresponding icosohedron.

The shell structure of this conic form of polyhedron is modularized in such a way as to build each vertex section 32 with five modules or structural members 33 to form an 'assembly 34 defining a frustoconical vertex section 32 and a portion of each of the five frustoconical facial sections 31 therearound. The complete conic icosohedral polyhedron approximating a spherical structure can be formed with twleve such vertex assemblies 34 or sixty identical modules in all.

As best seen in FIG. 6 each of the structural members 33 includes first and

second leg members

35 and 36, respectively, extending along edges of the corresponding icosohedral configuration, an outside arcuate portion 37 extending between

legs

35 and 36 and corresponding to a portion of the frustoconical facial section 31, and an inside arcuate portion 38 extending between

legs

35 and 36 preferably tangent to arcuate portions 37 and corresponding to a portion of the frustoconical vertex section 32.

Except for several of the vertical

conical sections

31 and 32 which are reserved for egress, ventilation and mounting of the building, the

frustoconical sections

31 and 32 are closed by

wall portions

41 and 42, respectively. As illustrated, each of these wall sections is formed by a pair of

disc members

43 and 44, such as of Mylar, coaxially mounted in the frustoconical section at their periphery by a rim or bead 45 easily insertible into a special inflated closure molding 46. A valve member 47 is provided for each pair of

disc members

43 and 44 for pressuring the space therebetween thereby giving the assembly a lens shape and shrinking the conical portions of the module members to which it is mounted together with a large pulling force.

Appropriate adjustment of the gas fill in the lens structure can render the building soundproof and metallization Y of the outside disc renders the lens transparent-to-translucent looking out of the structure but opaque looking in, as well as totally reflective looking both ways with regard to heat. This structure provides excellent vibrational damping. Where snow or wind load of a very large order is anticipated, lens systems can be stacked with inter-lens pressure so as to load share the applied forces.

The structural members 33 can include reinforcing wedges or be plastic foam filled as illustrated in FIG. 6

and connection ropes or cables 49 can be provided along the

arcuate portions

37 and 38 and/or

leg members

35 and 36 for attaching the different initially formed assemblies 34 together.

For the building structure, all utilities; drains, etc. enter through and are cast in a concrete cone support pedestal 51 which supports and anchors the house through one of the frustoconical side or major sections 31. This construction permits use of rough and hilly building sites, and numerous interesting division arrangements are possible.

In accordance with one embodiment of this invention, a primary floor platform 52 is provided resting on the support pedestal 51 and extending to the periphery of the polyhedral structure at that level. The floor platform can be made up of a plurality of wedge-shaped members 53, containing the remaining electricalconnections and duct work, which can be assembled on the support pedestal 51 and banded together by a peripheral holding cable 54. These elements 53 can include plug-in portions on their surface fo partitioning thereabove in a wide variety of patterns. The interior plumbing tree, wiring, ducts, etc. are fabricated in-plant and mounted on fiberboard sections which plug together at the building site to become concrete forms for pedestal 51.

For assembly of the structure the support pedestal 51 is provided with the frustoconical portion of one major or facial section 31 and five structural members 33 secured thereto to form-the first vertex assembly 34. The connection cables 49 can be threaded on site or prethreaded and connected to form twelve vertex assemblies 34'of five structural members 33 each. The shell can then be erected vertex-by-vertex by the same technique. When the entire shell is up, the

disc members

43 and 44 forming the

wall portions

41 and 42 can be inserted and pressurized in one rapid, almost tool-less operation. If the modules are not bonded at the time of assembly, the entire structure can be disassembled and re-erected at a different location.

By way of example, one housing unit built in accordance with this construction has a primary or platform diameter of 36-40 feet, an overall diameter of45 feet, frustoconical major, facial or side sections of 13 feet in diameter, frustoconical minor or vertex sections of IO feet in diameter, a length for the structural members 33 of about feet and a total weight for an empty house of 3-4 tons.

If it is desired to further modularize the structure shown in FIG. 6, the structural members 33 can be made up of left hand and right hand, half

structural members

56 and 57 as illustrated in FIG. 7.

As illustrated in FIG. 9, the inflated closure molding 46 can include a first substantially rigid retaining member 81 in the form of a bead extending around the frustoconical section 42 and provided with a groove 82 in one surface thereof. Another support, partially flexible ridge 83 provided with a groove 84 facing groove 82 is provided adjacent retaining member 81 but with a slot opening into the chamber defined by

grooves

82 and 83 adapted to receive the bead 45 at the periphery of the

discs

43 and 44. The ridge 83 includes internally thereof an inflatable chamber 85 which forces the ridge 83 against retaining member 81 to clamp the bead 45 in the

grooves

82 and 84.

While FIG. 8 illustrated the manner in which the

discs

43 and 44 can be positioned in the frustoconical section and the space 40 therebetween pressurized to hold the portions of the frustoconical sections together, other constructions of the disc members are possible. For example, with reference to FIG. 10, the discs 43' and 44 can be mounted at spaced apart locations adjacent the larger and smaller ends of the frustoconical section 42. Then the space 40 between the discs 43' and 44' can be partially evacuated through a valve member 47' whereby the stretching force on the disc members due to the difference in pressure on opposite sides thereof will hold the portions of the conical section together and serve to support the integrity of the overall structure.

Referring now to FIG. 11, there is shown still another embodiment of the present invention using a plurality of discs wherein a pair of discs 43a and 44a are mounted along a common line on the surface of the frustoconical section 42" and a third disc 44b mounted along a line spaced from the, line for mounting discs 43a and 43b. The space 40a between discs 43a and 44a can be pressurized while the space 40b between discs 44a and 44b can be partiallyevacuated to give both desired rigidity to the overall structure and accomplish multiple functions with the different discs and spaces be tween the'discs for producing desired thermal, acoustic and optical characteristics for the structure formed of a plurality of these frustoconical sections.

It will be appreciated that frustoconical sections forming either the sides-of a polyhedron orlocated at the vertices of the polyhedron can, where desired, be subdivided into other frustoconical sections of appropriate dimension.

As will be appreciated from the description above, one aspect of the present invention for providing an easily assembled structure from a plurality of identical modules is the provision of a polyhedral structure that is modular at the vertex. This concept is applicable not only to the conic forms of polyhedra, such as those illustrated in FIGS. 1-5, but also to regular polyhedra. By way of example, reference is now made'to, FIG. 12 showing the edge members of a dodecahedron formed by support members 62 which include edge portions 63 and 64'extending from a vertex 65 of the structure along two edges to two of the adjacent vertices. By assembling the same number of these support members 62 as there are edges meeting at the vertex 65, the vertex of the polyhedron 61 is established. Next, by assembling the required number of vertices together, the entire polyhedron is formed. While it is preferred that each of the

leg members

63 and 64 extend half way from the vertex 65 to an adjacent vertex for joinder with leg members from such adjacent vertex, these leg members can be of a different length and the shorter or

longer leg members

63 and 64 connected by connection members.

The transformation of the assembly modular members shown in FIG. 12 for the formation of a conic form of polyhedron is illustrated in FIG. 13 wherein an arcuate portion 68, such as the arcuate portions in the module of preceding figures, extends between

leg members

63 and 64. Naturally, advantages of the conic form of pol'yhedra such as strength can be sacrificed and hybrid structures, such as oval rather than circular configurations, utilized. As illustrated in FIG. 14, a polyhedral structure in accordance with the present invention can be formed utilizing oval rather than conic portions for the modular members and wherein left and

right hand modules

71 and 72 are used.

Obviously certain modifications in the structures and structural members described above can be made within the scope of the present invention. For example, while the invention has been illustrated in FIGS. lwith respect to regular polyhedra, other geometrical shapes can be used such as nonregular structures or partial nonregular structures.

A structure can be formed of different dimensioned modular vertex elements such as the example shown in FIG. 15. As shown there a rhombic dodecahedron can be formed with elliptical facial or side sections wherein modular elements 81 form the longer arcuate sides of a conic elliptical section for the rhombic dodecahedron vertices formed by three edges and modular elements 82 form the shorter arcuate sides of the conic elliptical section for the rhombic dodecahedron vertices formed by four edges.

Even though there is one origin or geometric center of the structure illustrated and described thus far, hybrid structures with more than one origin are possible constructions of the modular vertex system. Similarly, considering the distance from the origin or origins to the arcuate portions of the frustoconical sections as the radius, a single structure can be formed with conic section of different radii or varying radii such as shown in FIG. 15.

What is claimed is:

1. A polyhedral structure having a plurality of vertices and a plurality of edges meeting at each of the vertices and defining the faces of the polyhedral structure comprising:

a plurality of support frame members defining the edges which define at least certain faces of thepolyhedral structure, each of said support frame members having at least one leg member of a pair of leg members extending at least part way along edges defining such faces and at least one curved portion defining an arc of a substantially circular area centered in the face of the I polyhedral structure defined by such edges, the plurality of support frame members at each face joined together with the curved portions thereof completely surrounding said substantially circular area thereof, I at least a pair of substantially circular membrane wall members closing said area within said surrounding support frame members at each of such faces, means for coaxially mounting each pair of said membrane wall members from said curved portions of said surrounding support frame members completely around the periphery of said area and sealing the space between each pair of said membrane wall members, and v means for adjusting the pressure within the space between each pair of said membrane wall members causing said membrane wall members to tension said surrounding support frame members radially inward of said area. 2. The structure of claim '1 wherein the space between the membrane members of at least certain pairs of membrane'wall members is evacuated to be below atmospheric pressure.

3. The structure of claim 1 wherein the space between the membrane members of at least certain pairs of membrane wall members is pressurized to be above atmospheric pressure.

4. The polyhedral structure of claim 1 wherein said membrane wall members are plastic and at least one member of each pair is partially metalized.

5. The structure of claim 1 wherein at least certain membrane wall member pairs include a third membrane wall member, the space between the pair of membrane wall members being evacuated to be below atmospheric pressure and the space between said third membrane wall member and one of said pair of membrane wall members is pressurized to be above atmospheric pressure.

6. A structure comprising:

means defining a plurality of hollow frusto, substantially conical sections joined together with adjacent sections tangent to one another and the axes of all said sections intersecting in a point,

said section-defining means including a plurality of structural frame members,

each of said structural frame members having at least one leg member of a pair of leg members extending at least part way along tangent lines between sections and at least one curved portion defining an arc of a said conical section,

the plurality of frame members of each section joined together with the curved portions thereof completely surrounding said conical sections,

at least a pair of circular membrane wall members closing each conical section,

means for coaxially mounting each pair of said membrane wall members from said curved portions of said surrounding frame members completely around the periphery of said conical section and sealing the space between each pair of said membrane wall members, and

means for adjusting the pressure within the space between each pair of said membrane wall members causing said membrane wall members to tension said surrounding frame members radially inwardly of said conical sections.

7. The structure of claim 6 wherein the space between the membrane members of at least certain pairs of membrane wall members is evacuated to be below atmospheric pressure.

8. The structure of claim 6 wherein the space between the membrane members of at least certain pairs of membrane wall members is pressurized to be above atmospheric pressure;

9. The structure of claim 6 wherein said membrane wall members are plastic and at least one member of each pair is partially metalized.

10. The structure of claim 6 wherein at least certain membrane wall member pairs include a third membrane wall member, the space between the pair of membrane wall members being evacuated to be below atmospheric pressure and the space between said third membrane wall member and one of said pair of membrane wall members is pressurized to be above atmospheric pressure.

UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION patent 3,841,039 Dated October 15, 1974 Inventor) Philo T. Farnsworth II I It is certified that error appears in the above-identified patent and that said Letters Patent are hereby corrected as shown below:

The term of this patent subsequent to April 15, 1989, has been disclaixned.

Signed and sealed this 13th day of May 1975.

(SEAL) Attest:

C. MARSHALL DANN RUTH C. MASON Commissioner of Patents Attesting Officer and Trademarks FORM PO-\050(10-69) us covznnusnr Pmu'nuc orncs: 93 o