Polyhedral & Tessellation PISMP Semester 2 - LN

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<p>NUMERICAL LITERACY</p> <p>1.0</p> <p>ABSTRACT</p> <p>Tessellation and polyhedral are two types of art. Tessellations have been around for centuries and are still quite prevalent today. However the study of tessellations in mathematics has a relatively short history. In 1619, Johannes Kepler did one of the first documented studies of tessellations when he wrote about the regular and semi regular tessellation, which are coverings of a plane with regular polygons. Through history, polyhedral have been closely associated with the world of art. The peak of this relationship was certainly in the Renaissance. For some Renaissance artists, polyhedral simply provided challenging models to demonstrate their mastery of perspective. For others, polyhedral also were symbolic of deep religious or philosophical truths. For other artists, polyhedral simply provide inspiration and a storehouse of forms with various symmetries from which to draw on. This is especially so in twentieth century sculpture, free of the material and representational constraints of earlier conceptions of sculpture. In this assignment, we are going to provide two models of polyhedral and two designs of tessellation for each member. Besides that, we are ordered to provide the notes of both. In this report, we also will show the procedures or steps to create the models. The explanations comprised here are so helpful. Hopefully the whole contents of this particular assignment could help those deeply want to know about polyhedral and tessellation.</p> <p>KERJA KURSUS PENDEK</p> <p>1</p> <p>NUMERICAL LITERACY</p> <p>2.0</p> <p>POLYHEDRAL 2.1 NOTES OF POLYHEDRAL</p> <p>A polyhedron (plural polyhedra or polyhedrons) is often defined as a geometric solid with flat faces and straight edges. The word polyhedron has slightly different meanings in geometry and algebraic geometry. In geometry, a polyhedron is simply a three-dimensional solid which consists of a collection of polygons, usually joined at their edges. The word derives from the Greek poly (many) plus the Indo-European hedron (seat). A polyhedron is the three-dimensional version of the more general polytope (in the geometric sense), which can be defined in arbitrary dimension. The plural of polyhedron is "polyhedra" (or sometimes "polyhedrons").</p> <p>A convex polyhedron can be formally defined at the set of solutions to a system of linear inequalities.</p> <p>KERJA KURSUS PENDEK</p> <p>2</p> <p>NUMERICAL LITERACY</p> <p>2.1.1 DEFINITION Any polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions: 3 dimensions: The body is bounded by the faces, and is usually the volume enclosed by them.</p> <p>2 dimensions: A face is a polygon bounded by a circuit of edges, and usually including the flat (plane) region inside the boundary. These polygonal faces together make up the polyhedral surface.</p> <p>1 dimension: An edge joins one vertex to another and one face to another, and is usually a line segment. The edges together make up the polyhedral skeleton.</p> <p>0 dimensions: A vertex (plural vertices) is a corner point. -1 dimension: The nullity is a kind of non-entity required by abstract theories.</p> <p>A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. This ensures that the polyhedral surface is continuously connected and does not end abruptly or split off in different directions. A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions.</p> <p>KERJA KURSUS PENDEK</p> <p>3</p> <p>NUMERICAL LITERACY</p> <p>2.1.2 TYPES OF POLYHEDRAL There are three types of common polyhedral which are platonic solid or known as regular polyhedral, prisms and pyramids.</p> <p>2.1.1.1</p> <p>Platonic Solid or Regular Polyhedral</p> <p>There are only five regular polyhedral known to mathematics as the Platonic Solids which are; tetrahedron (four equilateral triangles); hexahedron (aka cube, six squares); octahedron (eight equilateral triangles); dodecahedron (12 pentagons); and icosahedrons (20 equilateral triangles).</p> <p>KERJA KURSUS PENDEK</p> <p>4</p> <p>NUMERICAL LITERACY</p> <p>Types of Platonic Solid Tetrahedron Cube Octahedron Dodecahedron Icosahedrons KERJA KURSUS PENDEK</p> <p>Characteristics</p> <p>Four faces Four vertices Six edges</p> <p>Six faces Eight vertices Twelve edges</p> <p>Eight faces Six vertices Twelve edges</p> <p>Twelve faces Twenty vertices Thirty edges</p> <p>Twenty faces Twelve vertices Thirty edges</p> <p>5</p> <p>NUMERICAL LITERACY</p> <p>As a conclusion, it will call Platonic Solids or Regular Polyhedral if it is vertextransitive, edge-transitive and face-transitive (this implies that every face is the same regular polygon; it also implies that every vertex is regular).</p> <p>2.1.1.2</p> <p>Prism</p> <p>A prism is also a polyhedron, which means the cross section will be polygon ( a straight-edged figure). A prism has the same cross section all along its length. That is mean, all sides will be flat. It has no curve sides. For example, a cylinder is not a prism. There are two kinds of prism which are regular prism and irregular prism. A regular prism is when the cross section is regular or in the other words, it is a shape with equal edge lengths. And the irregular prism is when the edge lengths of a shape not equal. A prism is named according to the shape of its base. Below are the examples of regular prism: 1) Regular Prism</p> <p>Types of Regular Prism Triangular Prism</p> <p>Cross-section</p> <p>KERJA KURSUS PENDEK</p> <p>6</p> <p>NUMERICAL LITERACY</p> <p>Square Prism</p> <p>Cube Prism</p> <p>Pentagonal Prism</p> <p>2) Irregular Prism</p> <p>KERJA KURSUS PENDEK</p> <p>7</p> <p>NUMERICAL LITERACY</p> <p>These models are called irregular prism because the pentagon is not regular in shape. But, the models are still known as prism and polyhedral.2.1.1.3</p> <p>Pyramid</p> <p>A particularly popular polyhedron is the pyramid. If we restrict ourselves to regular polygons for faces, there are three possible pyramids which are triangular pyramid, square pyramid and pentagonal pyramid. All the pyramids are named after the shape of their base.</p> <p>Types of Pyramids Triangular Pyramid</p> <p>Base</p> <p>Square Pyramid</p> <p>KERJA KURSUS PENDEK</p> <p>8</p> <p>NUMERICAL LITERACY</p> <p>Pentagonal Pyramid</p> <p>2.2 DESIGNS OF POLYHEDRAL</p> <p>In this assignment, we had created two types of polyhedral. For this part, we will show the procedures or steps to make the models and also a brief explanation about the both models. Besides, the procedure also have diagram and picture in order to simplify others to create these polyhedral too.</p> <p>2.2.1</p> <p>SPINNER (OCTAHEDRON) One of many wonderful designs by the late Lewis Simon, this model combines</p> <p>two standards bases, the Waterbomb base and the Preliminary base, to form a rigid modular construction. Use 12 sheets of fairly sturdy paper. The outer colour of the entire model will be the same as the outer colour of the Preliminary bases. 2.2.1.1 PROCEDURE</p> <p>KERJA KURSUS PENDEK</p> <p>9</p> <p>NUMERICAL LITERACY</p> <p>Step 1: Fold six Waterbomb and six Preliminary bases. Open out one of each base slightly, and allow the Preliminary base to wrap around the outside of the Waterbomb base, lining up the creases in the two bases.</p> <p>Step 2: Mountains fold each of the four corners of the Preliminary base inward, over the outer raw edges of the Waterbomb base, locking the sheets of paper together.</p> <p>Step 3: Allow the Waterbomb base to reform, two sheets folded as one. Repeat for remaining bases. the the</p> <p>KERJA KURSUS PENDEK</p> <p>10</p> <p>NUMERICAL LITERACY</p> <p>Step 4: Join any two units together by slipping the raw (Waterbomb) point of the first unit over the raw point of the other, but underneath the raw edge created by the Preliminary base. Push in all the way, until the two edges of what were the Preliminary bases meet.</p> <p>Step 5: In the same way, join adjacent points of the first two units into similar points of a third unit, forming a triangular section in the centre, as shown.</p> <p>Step 6: Add the final units in the same way the last unit is the most difficult to achieve. To activate the model place the points of the completed spinner into the centre of your palms, holding firmly. Blow hard against the top point of the model and the spinner will really spin.KERJA KURSUS PENDEK</p> <p>11</p> <p>NUMERICAL LITERACY</p> <p>2.2.1.2</p> <p>DESCRIPTION ABOUT THE MODEL</p> <p>For the first model, we had constructed the octahedron. To construct this model, we used 21cm x 21cm colour papers in order to construct the model. Besides, to construct this octahedron we need to cut 12 pieces of square paper and in order to construct this model, we made 6 units of module and join it together.</p> <p>KERJA KURSUS PENDEK</p> <p>12</p> <p>NUMERICAL LITERACY</p> <p>Actually, the model named as Spinner. It is because the model is easy to be spin. It also was functioning as a toy for kids. They love to spin and play with because it is easy to hold and not too hard. So, teachers can do origami for this model to use in the class. Besides, this model also suitable to be a decoration element. It is suit to be placed anywhere except in the toilet. The most appropriate place to be shown is on the table especially on the study table. It is because the pattern and shape are suitable to be laid on. And of course the colour itself can make the watcher feels happy due to the colouring style. So, hopefully this model is helpful to use in any condition as an art material. Octahedron</p> <p>See the octahedron rotating around the x, y, z axis.</p> <p>KERJA KURSUS PENDEK</p> <p>13</p> <p>NUMERICAL LITERACY</p> <p>Properties of the Octahedron Faces: 8 triangles Vertices: 6, each with 4 edges meeting Edges: 12 Dihedral angle: 10928'</p> <p>The Symmetry</p> <p>Surface Area</p> <p>KERJA KURSUS PENDEK</p> <p>14</p> <p>NUMERICAL LITERACY</p> <p>Let r = the distance from centres to one vertex. The length (a) of edge, by the Pythagoras Theorem, = r2.</p> <p>Then the area of one triangle is (a h) / 2, where h = [a - (a/2)]. And the area of the octahedron is 8 the area of one triangle.</p> <p>Volume</p> <p>The octahedron can be divided into two pyramids.</p> <p>The volume of one pyramid = (base area height) /3. In the case of the regular</p> <p>KERJA KURSUS PENDEK</p> <p>15</p> <p>NUMERICAL LITERACY</p> <p>octahedron, the base area = a. And so, the volume of the octahedron = 2 the volume of pyramid. V = (2 / 3)a Introduction to regular octahedron: In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is shown in the below figure.</p> <p>The properties and Cartesian coordinates of a regular octahedron are given below in detail. (Source: Wikipedia) Properties and Cartesian of Regular Octahedron:</p> <p>Number of faces in an octahedron = 8. i.e., it has 8 equilateral triangular faces.</p> <p>Number of vertices in an octahedron = 6</p> <p>KERJA KURSUS PENDEK</p> <p>16</p> <p>NUMERICAL LITERACY</p> <p>Number of edges in an octahedron = 12 An octahedron is a regular convex deltahedron. An octahedron has a Schlafi symbol of {3, 4}</p> <p>In octahedron, the vertices has the cartesian coordinates of (1, 0, 0); (0, 1, 0); (0, 0, ). The area and volume of regular octahedron is explained below with examples which help you to learn more about regular octahedron. 2.2.2 JAPANESE BROCADE (HEXAHEDRON) This design, by Minako Ishibasi, makes attractive earrings, when folded from extremely small squares, say 4 x 4cm/ 11/2 x 11/2in. for a practice version, begin with six squares of fairly sturdy paper. This modular design is very original, featuring curved surfaces, which seem to create the illusion of rings encircling the central cube shape. Only one shows on the finished unit, so begin with this colour face down. Two squares each of three different colours can also be used, as here.</p> <p>2.2.2.1</p> <p>PROCEDURE</p> <p>Step 1:</p> <p>KERJA KURSUS PENDEK</p> <p>17</p> <p>NUMERICAL LITERACY</p> <p>Begin by folding the square in half in one direction, to establish the centre line. Fold upper and lower edges into meet this crease.</p> <p>Step 2: Fold the right vertical edges upward on a diagonal crease to lie along the upper edge. Fold the left vertical edge downward to lie on the lower edge.</p> <p>Step 3: Unfold the paper completely.</p> <p>Step 4: Fold all four corners inward to lie on the horizontal quarter creases. Two of these creases will already have been made.</p> <p>KERJA KURSUS PENDEK</p> <p>18</p> <p>NUMERICAL LITERACY</p> <p>Step 5: Hold the upper and lower edges into lie along the horizontal quarter creases.</p> <p>Step 6: Fold the lower right flap inward on existing crease, made in step 2.</p> <p>Step 7: Peel back the inner vertical edge of the flap folded in step 6, on an existing parallel crease, while at the same time folding up the lower edge, also on the existing crease. This perform on both a swivel and a squash fold.</p> <p>Step 8:KERJA KURSUS PENDEK</p> <p>19</p> <p>NUMERICAL LITERACY</p> <p>Step 7 completed.</p> <p>Step 9: Repeats step 6-7 for the top left flap.</p> <p>Step 10: Pull out the lower border, and allow the flap folded in the step 9 to tuck in behind it. Flatten the model once more.</p> <p>Step 11: Step 10 completed.</p> <p>KERJA KURSUS PENDEK</p> <p>20</p> <p>NUMERICAL LITERACY</p> <p>Step 12: Turn the model over, and fold each of sharp points to obtuse angles of the parallelogram, as shown. the</p> <p>Step 13: Allow the flaps folded in step 12 to be unfolded slightly, to rest at right angles to the central squares shape. Make five more identical units.</p> <p>KERJA KURSUS PENDEK</p> <p>21</p> <p>NUMERICAL LITERACY</p> <p>Step 14: To assemble, slide the point of any one unit under the central section of another unit as shown.</p> <p>Step 15: Continue by adding a third unit, piece.</p> <p>assembling the central cube piece by If holding with two units of three different colours, you should add unit of same colours opposite each other. All units are joined in the same way, all the around the model.</p> <p>the the way</p> <p>KERJA KURSUS PENDEK</p> <p>22</p> <p>NUMERICAL LITERACY</p> <p>Step 16: Under construction.</p> <p>Step 17: Assembly finished.</p> <p>Step 18: Finally, squeeze together the four that appear on each of the six of the central cube, allowing them project upward slightly and form circular bands around the model. flaps faces to the</p> <p>KERJA KURSUS PENDEK</p> <p>23</p> <p>NUMERICAL LITERACY</p> <p>2.2.2.2</p> <p>DESCRIPTION ABOUT THE MODEL</p> <p>The second model is the cube or hexahedron. To make this cube, almost any type of paper can be used. As long as the paper is square, it can be folded. But, in our cube design, we used colour paper. Actually, the name was given for the model is Japanese Brocade. We wonder how they got the name, but the most important thing is the shape of the models. Whether it is interesting to watch or vice versa. It shapes like cube and clearly can be seen as cube after done. And our cube also can transform to another shape. It looks like a flower when it is opened. Thats meant, the model have an ecstatic value on it. That is why we decided to choose it and felt happy after finished it.</p> <p>KERJA KURSUS PENDEK</p> <p>24</p> <p>NUMERICAL L...</p>