Pyramid Geometry P.1/22
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Build a Tetrahedral Kite
Aeronautics Research Mission Directorate Build a Tetrahedral Kite Suggested Grades: 8-12 Activity Overview Time: 90-120 minutes In this activity, you will build a tetrahedral kite from Materials household supplies. • 24 straws (8 inches or less) - NOTE: The straws need to be Steps straight and the same length. If only flexible straws are available, 1. Cut a length of yarn/string 4 feet long. then cut off the flexible portion. • Two or three large spools of 2. Take six straws and place them on a flat surface. cotton string or yarn (approximately 100 feet total) 3. Use your piece of string to join three straws • Scissors together in a triangular shape. On the side where • Hot glue gun and hot glue sticks the two strings are extending from it, one end • Ruler or dowel rod for kite bridle should be approximately 20 inches long, and the • Four pieces of tissue paper (24 x other should be approximately 4 inches long. 18 inches or larger) See Figure 1. • All-purpose glue stick Figure 1 4. Tie these two ends of the string tightly together. Make sure there is no room for the triangle to wiggle. 5. The three straws should form a tight triangle. 6. Cut another 4-inch piece of string. 7. Take one end of the 4-inch string, and tie that end to a corner of the triangle that does not have the string ends extending from it. Figure 2. 8. Add two more straws onto the longest piece of string. 9. Next, take the string that holds the two additional straws and tie it to the end of one of the 4-inch strings to make another tight triangle. -
THE GEOMETRY of PYRAMIDS One of the More Interesting Solid
THE GEOMETRY OF PYRAMIDS One of the more interesting solid structures which has fascinated individuals for thousands of years going all the way back to the ancient Egyptians is the pyramid. It is a structure in which one takes a closed curve in the x-y plane and connects straight lines between every point on this curve and a fixed point P above the centroid of the curve. Classical pyramids such as the structures at Giza have square bases and lateral sides close in form to equilateral triangles. When the closed curve becomes a circle one obtains a cone and this cone becomes a cylindrical rod when point P is moved to infinity. It is our purpose here to discuss the properties of all N sided pyramids including their volume and surface area using only elementary calculus and geometry. Our starting point will be the following sketch- The base represents a regular N sided polygon with side length ‘a’ . The angle between neighboring radial lines r (shown in red) connecting the polygon vertices with its centroid is θ=2π/N. From this it follows, by the law of cosines, that the length r=a/sqrt[2(1- cos(θ))] . The area of the iscosolis triangle of sides r-a-r is- a a 2 a 2 1 cos( ) A r 2 T 2 4 4 (1 cos( ) From this we have that the area of the N sided polygon and hence the pyramid base will be- 2 2 1 cos( ) Na A N base 2 4 1 cos( ) N 2 It readily follows from this result that a square base N=4 has area Abase=a and a hexagon 2 base N=6 yields Abase= 3sqrt(3)a /2. -
Math 366 Lecture Notes Section 11.4 – Geometry in Three Dimensions
Section 11-4 Math 366 Lecture Notes Section 11.4 – Geometry in Three Dimensions Simple Closed Surfaces A simple closed surface has exactly one interior, no holes, and is hollow. A sphere is the set of all points at a given distance from a given point, the center . A sphere is a simple closed surface. A solid is a simple closed surface with all interior points. (see p. 726) A polyhedron is a simple closed surface made up of polygonal regions, or faces . The vertices of the polygonal regions are the vertices of the polyhedron, and the sides of each polygonal region are the edges of the polyhedron. (see p. 726-727) A prism is a polyhedron in which two congruent faces lie in parallel planes and the other faces are bounded by parallelograms. The parallel faces of a prism are the bases of the prism. A prism is usually names after its bases. The faces other than the bases are the lateral faces of a prism. A right prism is one in which the lateral faces are all bounded by rectangles. An oblique prism is one in which some of the lateral faces are not bounded by rectangles. To draw a prism: 1) Draw one of the bases. 2) Draw vertical segments of equal length from each vertex. 3) Connect the bottom endpoints to form the second base. Use dashed segments for edges that cannot be seen. A pyramid is a polyhedron determined by a polygon and a point not in the plane of the polygon. The pyramid consists of the triangular regions determined by the point and each pair of consecutive vertices of the polygon and the polygonal region determined by the polygon. -
VOLUME of POLYHEDRA USING a TETRAHEDRON BREAKUP We
VOLUME OF POLYHEDRA USING A TETRAHEDRON BREAKUP We have shown in an earlier note that any two dimensional polygon of N sides may be broken up into N-2 triangles T by drawing N-3 lines L connecting every second vertex. Thus the irregular pentagon shown has N=5,T=3, and L=2- With this information, one is at once led to the question-“ How can the volume of any polyhedron in 3D be determined using a set of smaller 3D volume elements”. These smaller 3D eelements are likely to be tetrahedra . This leads one to the conjecture that – A polyhedron with more four faces can have its volume represented by the sum of a certain number of sub-tetrahedra. The volume of any tetrahedron is given by the scalar triple product |V1xV2∙V3|/6, where the three Vs are vector representations of the three edges of the tetrahedron emanating from the same vertex. Here is a picture of one of these tetrahedra- Note that the base area of such a tetrahedron is given by |V1xV2]/2. When this area is multiplied by 1/3 of the height related to the third vector one finds the volume of any tetrahedron given by- x1 y1 z1 (V1xV2 ) V3 Abs Vol = x y z 6 6 2 2 2 x3 y3 z3 , where x,y, and z are the vector components. The next question which arises is how many tetrahedra are required to completely fill a polyhedron? We can arrive at an answer by looking at several different examples. Starting with one of the simplest examples consider the double-tetrahedron shown- It is clear that the entire volume can be generated by two equal volume tetrahedra whose vertexes are placed at [0,0,sqrt(2/3)] and [0,0,-sqrt(2/3)]. -
The Mars Pentad Time Pyramids the Quantum Space Time Fractal Harmonic Codex the Pentagonal Pyramid
The Mars Pentad Time Pyramids The Quantum Space Time Fractal Harmonic Codex The Pentagonal Pyramid Abstract: Early in this author’s labors while attempting to create the original Mars Pentad Time Pyramids document and pyramid drawings, a failure was experienced trying to develop a pentagonal pyramid with some form of tetrahedral geometry. This pentagonal pyramid is now approached again and refined to this tetrahedral criteria, creating tetrahedral angles in the pentagonal pyramid, and pentagonal [54] degree angles in the pentagon base for the pyramid. In the process another fine pentagonal pyramid was developed with pure pentagonal geometries using the value for ancient Egyptian Pyramid Pi = [22 / 7] = [aPi]. Also used is standard modern Pi and modern Phi in one of the two pyramids shown. Introduction: Achieved are two Pentagonal Pyramids: One creates tetrahedral angle [54 .735~] in the Side Angle {not the Side Face Angle}, using a novel height of the value of tetrahedral angle [19 .47122061] / by [5], and then the reverse tetrahedral angle is accomplished with the same tetrahedral [19 .47122061] angle value as the height, but “Harmonic Codexed” to [1 .947122061]! This achievement of using the second height mentioned, proves aspects of the Quantum Space Time Fractal Harmonic Codex. Also used is Height = [2], which replicates the [36] and [54] degree angles in the pentagon base to the Side Angles of the Pentagonal Pyramid. I have come to understand that there is not a “perfect” pentagonal pyramid. No matter what mathematical constants or geometry values used, there will be a slight factor of error inherent in the designs trying to attain tetrahedra. -
Pentagonal Pyramid
Chapter 9 Surfaces and Solids Copyright © Cengage Learning. All rights reserved. Pyramids, Area, and 9.2 Volume Copyright © Cengage Learning. All rights reserved. Pyramids, Area, and Volume The solids (space figures) shown in Figure 9.14 below are pyramids. In Figure 9.14(a), point A is noncoplanar with square base BCDE. In Figure 9.14(b), F is noncoplanar with its base, GHJ. (a) (b) Figure 9.14 3 Pyramids, Area, and Volume In each space pyramid, the noncoplanar point is joined to each vertex as well as each point of the base. A solid pyramid results when the noncoplanar point is joined both to points on the polygon as well as to points in its interior. Point A is known as the vertex or apex of the square pyramid; likewise, point F is the vertex or apex of the triangular pyramid. The pyramid of Figure 9.14(b) has four triangular faces; for this reason, it is called a tetrahedron. 4 Pyramids, Area, and Volume The pyramid in Figure 9.15 is a pentagonal pyramid. It has vertex K, pentagon LMNPQ for its base, and lateral edges and Although K is called the vertex of the pyramid, there are actually six vertices: K, L, M, N, P, and Q. Figure 9.15 The sides of the base and are base edges. 5 Pyramids, Area, and Volume All lateral faces of a pyramid are triangles; KLM is one of the five lateral faces of the pentagonal pyramid. Including base LMNPQ, this pyramid has a total of six faces. The altitude of the pyramid, of length h, is the line segment from the vertex K perpendicular to the plane of the base. -
Unit 6 Visualising Solid Shapes(Final)
• 3D shapes/objects are those which do not lie completely in a plane. • 3D objects have different views from different positions. • A solid is a polyhedron if it is made up of only polygonal faces, the faces meet at edges which are line segments and the edges meet at a point called vertex. • Euler’s formula for any polyhedron is, F + V – E = 2 Where F stands for number of faces, V for number of vertices and E for number of edges. • Types of polyhedrons: (a) Convex polyhedron A convex polyhedron is one in which all faces make it convex. e.g. (1) (2) (3) (4) 12/04/18 (1) and (2) are convex polyhedrons whereas (3) and (4) are non convex polyhedron. (b) Regular polyhedra or platonic solids: A polyhedron is regular if its faces are congruent regular polygons and the same number of faces meet at each vertex. For example, a cube is a platonic solid because all six of its faces are congruent squares. There are five such solids– tetrahedron, cube, octahedron, dodecahedron and icosahedron. e.g. • A prism is a polyhedron whose bottom and top faces (known as bases) are congruent polygons and faces known as lateral faces are parallelograms (when the side faces are rectangles, the shape is known as right prism). • A pyramid is a polyhedron whose base is a polygon and lateral faces are triangles. • A map depicts the location of a particular object/place in relation to other objects/places. The front, top and side of a figure are shown. Use centimetre cubes to build the figure. -
Classroom Capsules
CLASSROOM CAPSULES EDITOR Michael K. Kinyon Indiana University South Bend South Bend, IN 46634 Classroom Capsules consists primarily of short notes (1–3 pages) that convey new mathematical in- sights and effective teaching strategies for college mathematics instruction. Please submit manuscripts prepared according to the guidelines on the inside front cover to the Editor, Michael K. Kinyon, Indiana University South Bend, South Bend, IN 46634. The Theorem of Cosines for Pyramids Alexander Kheyfits (alexander.kheyfi[email protected]), Bronx Community College, CUNY, Bronx, NY 10453 The classical three-millennia-old Pythagorean theorem is still irresistibly attractive for lovers of mathematics—see, for example, [1] or a recent survey [8] for its numer- ous old and new generalizations. It can be extended to 3-space in various ways. A straightforward extension states that the square of a diagonal of a right rectangular parallelepiped is equal to the sum of squares of its adjacent edges. A less trivial gener- alization is the following result, which can be found in good collections of geometric problems as well as in a popular calculus textbook [7, p. 858]: (PTT) The Pythagorean theorem for tetrahedra with a trirectangular ver- tex. In a tetrahedron with a trirectangular vertex, the square of the area of the op- posite face to this vertex is equal to the sum of the squares of the areas of three other faces. Consider a triangle with sides a, b, c and the angle C opposite side c. The classical Pythagorean theorem is a particular case of the Theorem or Law of Cosines, c2 = a2 + b2 − 2ab cos C, if we specify here C to be a right angle. -
A Triangle with Sides Lengths of a Rational Power of the Plastic Constant
A TRIANGLE WITH SIDES LENGTHS OF A RATIONAL POWER OF THE PLASTIC CONSTANT KOUICHI NAKAGAWA Abstract. Using the golden ratio φ, a triangle whose side length ratio can be expressed as p 2 1 : φ : φ represents a rightp triangle since the golden ratio has the property of φ = φ + 1 and therefore satisfies 12 + ( φ)2 = φ2. This triangle is called the Kepler triangle. As in the case of the Kepler triangle, in this study we determine triangles where the length of the three sides is expressed using only the constant obtained from the linear recurrence sequence (golden ratio, plastic constant, tribonacci constant and supergolden ratio). 1. Introduction The Fibonacci numbers are defined by the recurrence Fn+2 = Fn+1 + Fn with the initial values F0 = 0, F1 = 1, and the limit of the ratios of successive terms converges to p F 1 + 5 lim n = φ = ≈ 1:61803 ··· ; n!1 Fn−1 2 φ is called the golden ratio. One property of the golden ratio is φ2 = φ + 1; it is also an algebraic integer of degree 2. By the property of the golden ratio and the Pythagorean theorem, we have φ2 = φ + 1 () 12 + (pφ)2 = φ2: p Since the ratio of the lengths of the three sides is 1 : φ : φ, this triangle is a right triangle, called the Kepler triangle. In addition, the following are known as characteristic triangles. An isosceles triangle whose three side length ratios can be expressed as φ : φ : 1 is called a golden triangle. The three angles are 36° − 72° − 72°. -
Putting the Icosahedron Into the Octahedron
Forum Geometricorum Volume 17 (2017) 63–71. FORUM GEOM ISSN 1534-1178 Putting the Icosahedron into the Octahedron Paris Pamfilos Abstract. We compute the dimensions of a regular tetragonal pyramid, which allows a cut by a plane along a regular pentagon. In addition, we relate this construction to a simple construction of the icosahedron and make a conjecture on the impossibility to generalize such sections of regular pyramids. 1. Pentagonal sections The present discussion could be entitled Organizing calculations with Menelaos, and originates from a problem from the book of Sharygin [2, p. 147], in which it is required to construct a pentagonal section of a regular pyramid with quadrangular F J I D T L C K H E M a x A G B U V Figure 1. Pentagonal section of quadrangular pyramid basis. The basis of the pyramid is a square of side-length a and the pyramid is assumed to be regular, i.e. its apex F is located on the orthogonal at the center E of the square at a distance h = |EF| from it (See Figure 1). The exercise asks for the determination of the height h if we know that there is a section GHIJK of the pyramid by a plane which is a regular pentagon. The section is tacitly assumed to be symmetric with respect to the diagonal symmetry plane AF C of the pyramid and defined by a plane through the three points K, G, I. The first two, K and G, lying symmetric with respect to the diagonal AC of the square. -
The Kepler Triangle and Its Kin
Forum Forma, 35, 1–2, 2020 The Kepler Triangle and Its Kin Takeshi Sugimoto Kanagawa University, Kanagawa Ward, Yokohama 221-8686, Japan E-mail address: [email protected] (Received July 29, 2019; Accepted September 30, 2019) The Kepler triangle, also known as the golden right triangle, is the right triangle with its sides of ratios ‘1 : φ1/2 : φ,’ where φ denotes the golden ratio. Also known are the silver right triangle and the square-root- three φ right triangle. This study introduces the generalised golden right triangle, which have sides of lengths 1/2 n/2 1/2 closely related to φ and the Fibonacci numbers, Fn:‘(Fn−2) : φ : (Fn) φ’ for any natural number n. This formalism covers all the known φ-related right triangle, i.e., the Kepler triangle and its kin. As n tends to infinity, the ratios of the sides go to ‘φ−1 :51/4 : φ.’ Our model plays an important role in the classroom to study the golden ratio, the Fibonacci numbers and the Pythagorean theorem. Keywords: Golden Ratio, Fibonacci Sequence, Pythagorean Theorem 1. Introduction as unity. Hence the above-mentioned relation is rewritten The point that Herodotus (485–425BCE) writes about as follows. King Khufu’s pyramid in his Histories, vol. II, article 124 1 + φ : φ = φ :1, (Lohmer,¨ 2006), is ‘... For the making of the pyramid it- or self there passed a period of twenty years; and the pyramid φ2 = φ + 1. is square, each side measuring eight hundred feet, and the The solution is given by height of it is the same. -
The Format of the IJOPCM, First Submission
Journal of Advances in Physics vol 16 No 1 (2019) ISSN: 2347-3487 https://rajpub.com/index.php/jap DOI: https://doi.org/10.24297/jap.v16i1.8402 Fine-Structure Constant from Sommerfeld To Feynman Michael A. Sherbon Case Western Reserve University Alumnus, United States [email protected] Abstract The fine-structure constant, which determines the strength of the electromagnetic interaction, is briefly reviewed beginning with its introduction by Arnold Sommerfeld and also includes the interest of Wolfgang Pauli, Paul Dirac, Richard Feynman and others. Sommerfeld was very much a Pythagorean and sometimes compared to Johannes Kepler. The archetypal Pythagorean triangle has long been known as a hiding place for the golden ratio. More recently, the quartic polynomial has also been found as a hiding place for the golden ratio. The Kepler triangle, with its golden ratio proportions, is also a Pythagorean triangle. Combining classical harmonic proportions derived from Kepler’s triangle with quartic equations determine an approximate value for the fine- structure constant that is the same as that found in our previous work with the golden ratio geometry of the hydrogen atom. These results make further progress toward an understanding of the golden ratio as the basis for the fine-structure constant. Keywords: Fine-Structure Constant, Fundamental Constants, History Of Physics, Golden Ratio, Quartic Equa- tion. 1. Introduction Writing on the history of physics, Stephen Brush says that in 1916: Sommerfeld generalized Bohr’s model to include elliptical orbits in three dimensions. He treated the problem relativistically (using Einstein’s formula for the increase of mass with velocity), ...