McInerney, J., Chen, B.G.G., Theran, L., Santangelo, C.D., Rocklin, D.Z.: Hidden symmetries generate rigid folding mechanisms in periodic origami. MacKay, R.S.: Renormalisation in Area Preserving Maps. Ma, J., Feng, H., Chen, Y., Hou, D., You, Z., et al.: Folding of tubular waterbomb. American Society of Mechanical Engineers (2021) In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Imada, R., Tachi, T.: Geometry and kinematics of cylindrical waterbomb tessellation. thesis, Clermont Auvergne University (2018) Today 21(3), 241–264 (2018)įeng, H.: Kinematics of spatial linkages and its applications to rigid origami. KeywordsĬallens, S.J., Zadpoor, A.A.: From flat sheets to curved geometries: origami and kirigami approaches. Furthermore, by analyzing the mapping, we give proof of the conservation of the dynamical system. By changing parameters of the mappings and composite them, we generalize the dynamical system of waterbomb tube to that of various tubular origami tessellations and show their oscillating configurations. In this paper, we decompose the dynamical system of waterbomb tube into three steps and represent the one-step using the two kinds of mappings between zigzag polygonal linkages. Although the quasi-periodic behavior is the characteristic of conservative systems, whether the system is conservative has been unknown.
Recently, the authors reported that the kinematics of waterbomb tube depends on the discrete dynamical system that arises from the geometric constraints between modules and quasi-periodic solutions of the dynamical system generate oscillating configurations. The oscillation of tubular waterbomb tessellation is one example. There is no reflectional symmetry, nor is there rotational symmetry.Ī pentomino is the shape of five connected checkerboard squares.Folded surfaces of origami tessellations sometimes exhibit non-trivial behaviors, which have attracted much attention. In glide reflection, reflection and translation are used concurrently much like the following piece by Escher, Horseman. A rotation, or turn, occurs when an object is moved in a circular fashion around a central point which does not move.Ī good example of a rotation is one "wing" of a pinwheel which turns around the center point. Rotations always have a center, and an angle of rotation. Rotation is spinning the pattern around a point, rotating it. To reflect a shape across an axis is to plot a special corresponding point for every point in the original shape. If a reflection has been done correctly, you can draw an imaginary line right through the middle, and the two parts will be symmetrical "mirror" images. Most commonly flipped directly to the left or right (over a "y" axis) or flipped to the top or bottom (over an "x" axis), reflections can also be done at an angle. The translation shows the geometric shape in the same alignment as the original it does not turn or flip.Ī reflection is a shape that has been flipped. These were described by Escher.Ī translation is a shape that is simply translated, or slid, across the paper and drawn again in another place. There are 4 ways of moving a motif to another position in the pattern.
He adopted a highly mathematical approach with a systematic study using a notation which he invented himself. There are 17 possible ways that a pattern can be used to tile a flat surface or 'wallpaper'.Įscher read Pólya's 1924 paper on plane symmetry groups.Escher understood the 17 plane symmetry groups described in the mathematician Pólya's paper, even though he didn't understand the abstract concept of the groups discussed in the paper.īetween 19 Escher produced 43 colored drawings with a wide variety of symmetry types while working on possible periodic tilings. One mathematical idea that can be emphasized through tessellations is symmetry. If you look at a completed tessellation, you will see the original motif repeats in a pattern. The term has become more specialised and is often used to refer to pictures or tiles, mostly in the form of animals and other life forms, which cover the surface of a plane in a symmetrical way without overlapping or leaving gaps. They were used to make up 'tessellata' - the mosaic pictures forming floors and tilings in Roman buildings The word 'tessera' in latin means a small stone cube. When you fit individual tiles together with no gaps or overlaps to fill a flat space like a ceiling, wall, or floor, you have a tiling. A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps.Īnother word for a tessellation is a tiling.
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