BIAXIALLY TWISTED STRUCTURE
The present invention provides a biaxially twisted structure having curved developable surfaces which can be represented by a simple net. The net is in a two-dimensional shape of two congruent rotationally symmetric convex polygons misalignedly connected along a shared edge. A three-dimensional shape of the structure can be reconstructed from the net by sequentially connecting its edges. The structure has a unique design and mechanical feature, which can be utilized as a box, container, part, toy, building and so on.
The invention relates to a twisted structure having curved surfaces that can be represented by a simple net.
BACKGROUND OF THE INVENTIONA basic geometric shape of a typical box is a prism. In particular, a cuboid is practical since it can be represented by a simple net, the structure is stable, and it fills a space in three orthogonal directions. Other basic geometric shapes such as an antiprism are rarely used, and thus diversities are limited. Also, the shapes of boxes are mostly polyhedral, consisting of flat faces, while those having curved faces are few. Moreover, there is a limited way to create curved surfaces in boxes, primarily by changing a straight edge into a curved edge to create curved surfaces.
SUMMARY OF THE INVENTIONThe present invention, therefore, provides a novel structure with an attractive design and functional features that can be used as a box, container, toy, part, building and so on. A shape of the twisted structure is a curved body, twisted in two orthogonal axes, and can be represented by a simple net, which is different from common basic shapes like a prism or antiprism.
In one aspect, the present invention provides a structure, having a three-dimensional shape that can be represented by a net in a two-dimensional shape, the two-dimensional shape of the net including: 2 congruent rotationally symmetric convex polygons, wherein the polygon is a n-gon and n is an integer more than 3; 2n three-valent main-vertices, wherein each of the main-vertices is placed at each corner of the polygons without overlapping; 2n three-valent sub-vertices, wherein n of the sub-vertices are placed at rotationally symmetric positions within each of the polygons without overlapping; 2n main-edges, wherein each of the main-edges is straight, and each of the main-edges connects 2 of the main-vertices without crossing; and 4n sub-edges, wherein each of the sub-edges is either straight or curved, each of the sub-edges connects two of the sub-vertices or connects one of the main-vertices and one of the sub-vertices, and 2n of the sub-edges are placed at rotationally symmetric positions within each of the polygons without crossing, wherein: each of the main-vertices contacts 2 of the main-edges and one of the sub-edges, and each of the sub-vertices contacts 1 main-edge and the 2 sub-edges, and wherein the structure has curved developable surfaces with a biaxially twisted appearance. The net is simple but structure represented by the net is rather complex. The main-edges form a basic skeleton of the structure, and the sub-edges expand the degree of freedom in creating new designs. Incorporation of regular polygons in the net endows the structure with graceful appearance.
In another aspect, the present invention provides said structure, wherein the structure is a box and the net further includes a plurality of flaps and/or crease lines. A box of this new design can be made without using tools or other elements.
In a further aspect, the present invention provides said structure, wherein the structure rotates in a flow about its rotational axis when a direction of the rotational axis is in parallel to a direction of the flow. Providing a shaft, the structure rotates in a stream like a pinwheel.
The invention was conceived during analyses of a number of polyhedra with various basic geometric shapes. In particular, the idea was developed from a certain kind of polyhedra, consisting of only 4-gons (i.e. tetragons) with 3- and 4-valent vertices (e.g. “3-valent vertex” is a vertex with 3 edges), among those homeomorphic to a sphere. The structure of this invention, however, is not a polyhedron, but a curved body made of developable surfaces (“developable surface” is a surface that can be flattened onto a plane without stretching or compressing).
The structure described herein is complex but it can be represented by a simple net (“net” is a two-dimensional shape obtained by flattening a three-dimensional shape). It has a basic geometric shape different from a prism or antiprism. The structure is twisted in two orthogonal axes and may provide an elegant exterior. The top and a bottom faces are near-flat, which may be stably placed on a level surface. A square, rectangular, or hexagonal type resembles a space-filling solid, and is suitable for storing. Furthermore, the structure has unique features, in which it rotates in a stream or displays awkward motions under defined conditions.
This structure may look similar to an antiprism, but these are intrinsically different. The antiprism is a polyhedron: 1) all edges are straight, 2) all faces are planar, 3) all vertices are 4-valent, 4) the twist is one-axial (horizontal axis), and 5) side faces are triangles. The twisted structure, on the other hand, is a curved body: 1) edges are curved, 2) the faces are curved, 3) the vertices are basically 3-valent, 4) the twist is bi-axial (horizontal and vertical axes), and 5) side faces are pentagons (tetragon-like pentagons, in some cases). However, as described later, an antiprism may be obtained as an ultimate shape upon transformation of the twisted structure.
The net of the twisted structure has a shape of two congruent rotationally symmetric convex polygons which are connected along a shared edge with a parallel displacement along the shared edge (call it “misalignedly connected”). Here, “congruent” means a state of polygons having the same shape and size that perfectly overlap each other. “rotational symmetry” refers to symmetry that the original figure perfectly overlaps with that rotated by an angle of 360/n degrees (n-fold symmetry, wherein n is an integer greater than or equal to 2) about the rotation axis. A “convex polygon” is a polygon with every interior angle of less than 180 degrees and each edge being a straight line segment. Taking the net of the square twisted structure as an example, the shape of the polygon is a square, which is convex (every internal angle is 90 degrees) and rotationally symmetric (4-fold symmetry), and two congruent squares are misalignedly connected along the shared edge. This looks simpler than a typical net of a prism.
There are several ways to misalignedly connect two congruent convex polygons.
As shown in
The twisted structure can be designed based on other convex polygons.
In addition to the main-vertices, 2n secondary vertices (sub-vertices) may be chosen at rotationally symmetric positions within the convex polygons in the net. Sub-vertices expand the degree of freedom in designing. In
Taking the square twisted structure as an example, the approximate height H and twist angle (each for the top or bottom center area) ε are calculated. To simplify, sub-edges are made in parallel to main-edges. A is the offset width of two squares and B is the width of the center area.
Rectangular twisted structures is complex. The center area is approximately flat, but curvature in the diagonal direction is more evident depending on aspect ratios. Heights at the long and short sides are different, where the long side is higher, making it less stable as compared to the square-type. However, it creates unique motion including swaying back and forth when placed on a level surface.
A transformation of the twisted structure ultimately leads to an antiprism. The transformation is achieved by shifting the position of each sub-vertex to the main-vertex connected (e.g. a->2, b->4, c->6, and d->8 for the top face, e->1, f->3, g->5, and h->7 for the bottom face in
All the twisted structure as shown in
The twisted structure has curved surfaces. When materials such as a paper and plastic sheet are used (substantially flat and flexible, but relatively hard to stretch), structural strains will be generated as curved surfaces are formed. Such strains might destabilize curved structures as compared to polyhedral structure. However, necessary measures known in the art (such as adhesion and hooking) provide sufficient strength.
For other materials, the twisted structure are produced only to have a “twisted appearance” without introducing unnecessary strains. In this case, three-dimensional data are useful, which are obtained by computer modeling (e.g. CAD modeling or theoretical calculation based on the net) or actual modeling (e.g. 3D scanning of a prototype obtained from the net). For production, various methods known in the art may be employed (molding, 3D printing, NC processing, etc.). For example, containers may be produced by molding (e.g. injection molding of plastic resins), and parts may be produced by 3D printing or NC processing. For buildings, skeletal structures described herein are used as primary frames, which may be strengthened with secondary frames by triangulating the primary frames.
A regular-polygonal twisted structure resembles a pinwheel or propeller. In fact, providing a shaft to the axis of rotation, the structure turns as you blow on.
Another example regarding motion is provided in
Claims
1. A structure, having a three-dimensional shape that can be represented by a net in a two-dimensional shape, the two-dimensional shape of the net comprising:
- 2 congruent rotationally symmetric convex polygons, wherein the polygon is a n-gon and n is an integer more than 3;
- 2n three-valent main-vertices, wherein each of the main-vertices is placed at each corner of the polygons without overlapping;
- 2n three-valent sub-vertices, wherein n of the sub-vertices are placed at rotationally symmetric positions within each of the polygons without overlapping;
- 2n main-edges, wherein each of the main-edges is straight, and each of the main-edges connects 2 of the main-vertices without crossing; and
- 4n sub-edges, wherein each of the sub-edges is either straight or curved, each of the sub-edges connects two of the sub-vertices or connects one of the main-vertices and one of the sub-vertices, and 2n of the sub-edges are placed at rotationally symmetric positions within each of the polygons without crossing,
- wherein:
- each of the main-vertices contacts 2 of the main-edges and one of the sub-edges, and
- each of the sub-vertices contacts 1 main-edge and the 2 sub-edges, and
- wherein the structure has curved developable surfaces with a biaxially twisted appearance.
2. The structure of claim 1, wherein the polygon is a regular polygon.
3. The structure of claim 1, wherein the structure is a box and the net further comprises a plurality of flaps and/or crease lines.
4. The structure of claim 1, wherein the structure is a container formed in the three-dimensional shape that can be represented by the net.
5. The structure of claim 1, wherein the structure rotates in a flow about its rotational axis when a direction of the rotational axis is in parallel to a direction of the flow.
Type: Application
Filed: May 10, 2020
Publication Date: Dec 21, 2023
Inventors: Hiroshi Udo (Fukuoka), Takako Udo (Fukuoka)
Application Number: 16/871,034