1970-2019 TOPIC INDEX for the College Mathematics Journal (Including the Two Year College Mathematics Journal)
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Shape Skeletons Creating Polyhedra with Straws
Shape Skeletons Creating Polyhedra with Straws Topics: 3-Dimensional Shapes, Regular Solids, Geometry Materials List Drinking straws or stir straws, cut in Use simple materials to investigate regular or advanced 3-dimensional shapes. half Fun to create, these shapes make wonderful showpieces and learning tools! Paperclips to use with the drinking Assembly straws or chenille 1. Choose which shape to construct. Note: the 4-sided tetrahedron, 8-sided stems to use with octahedron, and 20-sided icosahedron have triangular faces and will form sturdier the stir straws skeletal shapes. The 6-sided cube with square faces and the 12-sided Scissors dodecahedron with pentagonal faces will be less sturdy. See the Taking it Appropriate tool for Further section. cutting the wire in the chenille stems, Platonic Solids if used This activity can be used to teach: Common Core Math Tetrahedron Cube Octahedron Dodecahedron Icosahedron Standards: Angles and volume Polyhedron Faces Shape of Face Edges Vertices and measurement Tetrahedron 4 Triangles 6 4 (Measurement & Cube 6 Squares 12 8 Data, Grade 4, 5, 6, & Octahedron 8 Triangles 12 6 7; Grade 5, 3, 4, & 5) Dodecahedron 12 Pentagons 30 20 2-Dimensional and 3- Dimensional Shapes Icosahedron 20 Triangles 30 12 (Geometry, Grades 2- 12) 2. Use the table and images above to construct the selected shape by creating one or Problem Solving and more face shapes and then add straws or join shapes at each of the vertices: Reasoning a. For drinking straws and paperclips: Bend the (Mathematical paperclips so that the 2 loops form a “V” or “L” Practices Grades 2- shape as needed, widen the narrower loop and insert 12) one loop into the end of one straw half, and the other loop into another straw half. -
National Forests Non-Motorized Trail Strategy the US Forest Service Is Looking for the 82 People Who Attended
SECOND QUARTER 2012 Quarterly News Bulletin and Hike Schedule P.O. Box 68, Asheville, NC 28802 • www.carolinamountainclub.org • e-mail: [email protected] National Forests non-motorized trail strategy The US Forest service is looking for the 82 people who attended. Some assistance. I wonder, though, if the group a few good men and women. Since the thoughts by CMC attendees: (bikers, horsemen, hunters, hikers, etc.) that majority of trail work is done by vol- Tish Desjardins, CMC Chair of provides the best grant assistance ends up get- unteers, the Forest Service is conduct- Conservation, said: “I thought it was ting what they want in the forest. With all the ing a number of workshops at various interesting that they are looking to us maintenance that CMC performs, it would be locations in Western North Carolina for possibly applying for grants for destructive if bikes or horses come along on to bring different types of trail users projects that we could apply for. They the hiking trails that we maintain. Hopefully together to provide input on trail plan- sure seem to be desperate for financial continued on page 7 ning. The diverse types of trail users include hikers, bikers, and horsemen. These people were brought together to share trail experiences, identify the types of trails that fit in a recreational context, and help develop a sustainable system of trail management. Initial workshops were held in Morganton, Andrews, Mars Hill, Franklin and Brevard. Several CMC members joined the 264 participants in the five meetings. Seventy diverse organizations were represented. The workshop in Brevard had to be moved to a larger location to accommodate Trail strategy participants at the meeting in Franklin. -
Polygrams and Polygons
Poligrams and Polygons THE TRIANGLE The Triangle is the only Lineal Figure into which all surfaces can be reduced, for every Polygon can be divided into Triangles by drawing lines from its angles to its centre. Thus the Triangle is the first and simplest of all Lineal Figures. We refer to the Triad operating in all things, to the 3 Supernal Sephiroth, and to Binah the 3rd Sephirah. Among the Planets it is especially referred to Saturn; and among the Elements to Fire. As the colour of Saturn is black and the Triangle that of Fire, the Black Triangle will represent Saturn, and the Red Fire. The 3 Angles also symbolize the 3 Alchemical Principles of Nature, Mercury, Sulphur, and Salt. As there are 3600 in every great circle, the number of degrees cut off between its angles when inscribed within a Circle will be 120°, the number forming the astrological Trine inscribing the Trine within a circle, that is, reflected from every second point. THE SQUARE The Square is an important lineal figure which naturally represents stability and equilibrium. It includes the idea of surface and superficial measurement. It refers to the Quaternary in all things and to the Tetrad of the Letter of the Holy Name Tetragrammaton operating through the four Elements of Fire, Water, Air, and Earth. It is allotted to Chesed, the 4th Sephirah, and among the Planets it is referred to Jupiter. As representing the 4 Elements it represents their ultimation with the material form. The 4 angles also include the ideas of the 2 extremities of the Horizon, and the 2 extremities of the Median, which latter are usually called the Zenith and the Nadir: also the 4 Cardinal Points. -
Symmetry Is a Manifestation of Structural Harmony and Transformations of Geometric Structures, and Lies at the Very Foundation
Bridges Finland Conference Proceedings Some Girihs and Puzzles from the Interlocks of Similar or Complementary Figures Treatise Reza Sarhangi Department of Mathematics Towson University Towson, MD 21252 E-mail: [email protected] Abstract This paper is the second one to appear in the Bridges Proceedings that addresses some problems recorded in the Interlocks of Similar or Complementary Figures treatise. Most problems in the treatise are sketchy and some of them are incomprehensible. Nevertheless, this is the only document remaining from the medieval Persian that demonstrates how a girih can be constructed using compass and straightedge. Moreover, the treatise includes some puzzles in the transformation of a polygon into another one using mathematical formulas or dissection methods. It is believed that the document was written sometime between the 13th and 15th centuries by an anonymous mathematician/craftsman. The main intent of the present paper is to analyze a group of problems in this treatise to respond to questions such as what was in the mind of the treatise’s author, how the diagrams were constructed, is the conclusion offered by the author mathematically provable or is incorrect. All images, except for photographs, have been created by author. 1. Introduction There are a few documents such as treatises and scrolls in Persian mosaic design, that have survived for centuries. The Interlocks of Similar or Complementary Figures treatise [1], is one that is the source for this article. In the Interlocks document, one may find many interesting girihs and also some puzzles that are solved using mathematical formulas or dissection methods. Dissection, in the present literature, refers to cutting a geometric, two-dimensional shape, into pieces that can be rearranged to compose a different shape. -
Pisgah Forest, NC, 28769
OFFERING MEMORANDUM 3578 HENDERSONVILLE HWY | PISGAH FOREST, NC REPRESENTATIVE PHOTO ™ 3578 HENDERSONVILLE HWY | PISGAH FOREST, NC 3 INVESTMENT SUMMARY EXCLUSIVELY LISTED BY: WESLEY CONNOLLY Associate VIce President 4 D: +1 (949) 432-4512 FINANCIAL SUMMARY M: +1 (707) 477-7185 [email protected] License No. 01962332 (CA) 6 KYLE MATTHEWS Broker of Record TENANT PROFILE License No. C27092 (NC) 7 AREA OVERVIEW 2 Dollar General INVESTMENT SUMMARY 3578 Hendersonville Hwy ADDRESS Pisgah Forest, NC 28769 $1,360,780 6.15% $83,688 ±9,100 SF 2017 LIST PRICE CAP RATE ANNUAL RENT GLA YEAR BUILT PRICE $1,360,780 CAP RATE 6.15% NOI $83,688 INVESTMENT HIGHLIGHTS GLA ±9,100 SF Corporate Guaranteed Essential Retailer LOT SIZE ±6.72 AC • Newer construction building with long term absolute NNN Lease; No YEAR BUILT 2017 Landlord Responsibilities • Dollar General has investment grade rated corporate guarantee • Dollar General has been identified as an essential retailer and has maintained business operations throughout the Covid-19 Pandemic DEMOGRAPHICS Prototypical Dollar General Market 3-MILE 5-MILE 10-MILE • Lack of Major competition in immediate vicinity POPULATION 5,225 17,083 61,536 • 33 miles from Asheville, NC HOUSEHOLDS 2,453 7,440 27,369 • 10 Mile Population in excess of 61,615 HH INCOME $68,333 $73,778 $79,155 • Minutes to John Rock, Looking Glass Rock, and Coontree Mountain Dollar General 3 FINANCIAL SUMMARY ANNUALIZED OPERATING DATA LEASE COMMENCE MONTHLY RENT ANNUAL RENT CAP RATE Lease Type NNN Type of Ownership Fee Simple Current -
Pisgah District Trails
PISGAH RANGER DISTRICT TRAILS Table of Contents: Trail Name Length Rating Trail Use Page # Andy Cove Nature Trail 0.7mi Easy Hiking 3 Art Loeb Spur 0.6mi Difficult Hiking 4 Art Loeb Trail–Section 1 12.3mi Difficult Hiking 5 Art Loeb Trail-Section 2 7.2mi Difficult Hiking 6 Art Loeb Trail-Section 3 6.8mi Difficult Hiking 7 Art Loeb Trail-Section 4 3.8mi Difficult Hiking 8 Avery Creek 3.2mi Medium Hiking/Biking 9 Bad Fork 2.0mi Difficult Hiking 10 Bennett Gap 2.9mi Medium Hiking/Biking(seasonal) 11 Big Creek 4.9mi Difficult Hiking/Sections open to bikes& horses 12 Biltmore Campus 0.9mi Easy Hiking/ wheelchair accessible 13 Black Mountain 9.8mi Difficult Hiking/Biking 14 Boyd Branch 0.7mi Easy Hiking/Biking 15 Bradley Creek 5.1mi Medium Hiking/Biking/Horses 16 Buck Spring 6.2mi Medium Hiking 17 Buckeye Gap 3.1mi Difficult Hiking 18 Buckhorn Gap 5.2mi Medium Hiking/Biking/Horses 19 Buckwheat Knob 1.5mi Medium Hiking/Biking 20 Butter Gap 3.4mi Medium Hiking/Biking 21 Caney Bottom 4.6mi Medium Hiking/Sections open to biking 22 Case Camp Ridge 1.7mi Difficult Hiking 23 Cat Gap Bypass 0.4mi Easy Hiking 24 Cat Gap Loop 4.4mi Medium Hiking/Sections open to biking(seasonal) 25 Chestnut Cove 0.2mi Medium Hiking 26 Clawhammer Cove 1.5mi Medium Hiking 27 Club Gap 0.8mi Medium Hiking/Biking 28 Cold Mountain 1.4mi Difficult Hiking 29 Coontree Loop 3.7mi Medium Hiking/Sections open to biking(seasonal) 30 Courthouse Falls 0.3mi Easy Hiking 31 Cove Creek 0.7mi Easy Hiking/Biking 32 Daniel Ridge Loop 4.0mi Medium Hiking/Biking 33 Deer Lake Lodge 1.4mi Medium -
The Order in Creation in Number and Geometry
THE ORDER IN CREATION IN NUMBER AND GEOMETRY Tontyn Hopman www.adhikara.com/number_and_geometry.html Illustrations by the author 2 About the origin of “The Order in Creation in Number and Geometry”. The author, Frederik (Tontyn) Hopman, was born in Holland in 1914, where he studied to become an architect. At the age of 18, after the death of his father, he had a powerful experience that led to his subsequent study of Oriental esoteric teachings. This was to become a life-long fascination. Responding to the call of the East, at the age of 21, he travelled to India by car. In those days this adventurous journey took many weeks. Once in India he married his travelling companion and settled down in Kashmir, where he lived with his young family for 12 years until in 1947 the invasion from Pakistan forced them to flee. Still in Asia, at the age of 38, Tontyn Hopman had a profound Kundalini awakening that gave his life a new dimension. It was during this awakening that he had a vision of Genesis, which revealed to him the ‘Order in Creation in Number and Geometry’. Around this time, however, Tontyn Hopman decided to return to Europe to enable his children to have a good education and he settled in Switzerland to practise his profession as an architect. Later he occupied himself with astrology and art therapy. Here, in Switzerland, after almost half a century, the memory of his vision came up again, with great clarity. Tontyn Hopman experienced a strong impulse to work on, and present the images that had been dormant for such a long time to the wider public. -
Dissecting and Folding Stacked Geometric Figures
Dissecting and Folding Stacked Geometric Figures Greg N. Frederickson Purdue University Copyright notice: Permission is granted only to DIMACS to post this copy of Greg N. Frederickson’s talk, “Dissecting and Folding Stacked Geometric Figures”, on its website. No permission is granted for others to copy any portions of this talk for their own or anyone else’s use. If you have what you feel is a justifiable use of this talk, contact me ( [email protected] ) to ask permission. Because producing this material was a lot of work, I reserve the right to deny any particular request. 1 A note about animations and videos None of the animations and videos were included when my powerpoint file was converted to a pdf file. Any slide that contains a picture with color but no title is the first frame of an animation or video. Selected animations are available from my webpages: http://www.cs.purdue.edu/homes/gnf/book3/stackfold.html http://www.cs.purdue.edu/homes/gnf/book3/stackfold2.html These animations were time-intensive to produce. Please show your appreciation by respecting my copyright. Dissection of one square to two squares [Plato, 4 th century, BCE] 2 Hinged dissection of one square to two squares (with 2 swing-hinged assemblages) Piano-hinge (fold-hinge) brings a piece A from being next to a piece B to being on top of B 3 Piano-hinged dissection of 1 square to 2 squares (with 2 assemblages) 4 Juggling 2 (or more) assemblages is a pain! So let’s change the game: Instead of having two smaller squares, let’s have just one , but twice as high. -
Kabbalah, Magic & the Great Work of Self Transformation
KABBALAH, MAGIC AHD THE GREAT WORK Of SELf-TRAHSfORMATIOH A COMPL€T€ COURS€ LYAM THOMAS CHRISTOPHER Llewellyn Publications Woodbury, Minnesota Contents Acknowledgments Vl1 one Though Only a Few Will Rise 1 two The First Steps 15 three The Secret Lineage 35 four Neophyte 57 five That Darkly Splendid World 89 SIX The Mind Born of Matter 129 seven The Liquid Intelligence 175 eight Fuel for the Fire 227 ntne The Portal 267 ten The Work of the Adept 315 Appendix A: The Consecration ofthe Adeptus Wand 331 Appendix B: Suggested Forms ofExercise 345 Endnotes 353 Works Cited 359 Index 363 Acknowledgments The first challenge to appear before the new student of magic is the overwhehning amount of published material from which he must prepare a road map of self-initiation. Without guidance, this is usually impossible. Therefore, lowe my biggest thanks to Peter and Laura Yorke of Ra Horakhty Temple, who provided my first exposure to self-initiation techniques in the Golden Dawn. Their years of expe rience with the Golden Dawn material yielded a structure of carefully selected ex ercises, which their students still use today to bring about a gradual transformation. WIthout such well-prescribed use of the Golden Dawn's techniques, it would have been difficult to make progress in its grade system. The basic structure of the course in this book is built on a foundation of the Golden Dawn's elemental grade system as my teachers passed it on. In particular, it develops further their choice to use the color correspondences of the Four Worlds, a piece of the original Golden Dawn system that very few occultists have recognized as an ini tiatory tool. -
The Treasures of the Snow
THE TREASURES OF THE SNOW In what many claim to be the oldest Book of the Bible, the Lord God asks Job a question that should interest its readers: “Hast thou entered into the treasures of the snow…?” (Job 38:22). Unsurprisingly – the day of magnifying glass and microscope far distant – we gather that Job hadn’t – and thus infer that the question is really addressed to those more favourably placed. Indeed, we know that some will already have ‘entered into’ the wonders associated with a snow crystal – as ideally represented here…* …and will have been informed that of the myriad of such crystals that have fallen to earth since the Creation, no two are identical. So, we may reasonably ask, “What does the Designer intend us to learn from this beautiful, super-abundant, structure?” The beginnings of an answer must surely include the following: 6 is prominently displayed 1. in the outline of the central regular hexagon, and 2. in the overall form of the regular 6-pointed star, or hexagram. We therefore gather that six is highly regarded in the courts of heaven. But also from man’s point of view, it is the first of what mathematicians refer to as a perfect number [the term associated with any number that is equal to the sum of its factors – in this case, 1+2+3]. The sequence continues 28, 496, 8128…. – apparently without limit. Again, all known perfect numbers are triangular [meaning that they possess the geometry of an equilateral triangle when expressed absolutely as a close formation of uniform circular counters]. -
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[Math.AG] 3 Aug 2005
CURVES IN CAGES: AN ALGEBRO-GEOMETRIC ZOO GABRIEL KATZ 1. Introduction. An algebraic plane curve is the solution set of a polynomial equation P (x, y)=0, where x and y are real or complex variables. By definition, the degree of the curve is the degree of the polynomial P (x, y).1 When P (x, y) is a product of two non-constant polynomials over a given number field, the curve is called reducible; otherwise, it is irreducible. This paper is concerned with families of plane algebraic curves that contain a given and quite special set of points X . We focus on the case in which the set X is formed by transversally intersecting pairs of lines selected from two given finite families. The union of all lines from both families is called a cage (this notion of cage will be made more precise later), and the intersection X consists of points at which a line from the first family intersects a line from the second. The points of X are called the nodes of the cage. This is a particular case of a more general problem. Let X be the intersection set of two plane algebraic curves D and E that do not share a common component, that is, do not contain a common irreducible curve. If d and e denote the degrees of D and E, respectively, then X consists of at most d · e points. When the cardinality of X is exactly d · e, X is called a complete intersection (complete intersections have many nice properties). -
MYSTERIES of the EQUILATERAL TRIANGLE, First Published 2010
MYSTERIES OF THE EQUILATERAL TRIANGLE Brian J. McCartin Applied Mathematics Kettering University HIKARI LT D HIKARI LTD Hikari Ltd is a publisher of international scientific journals and books. www.m-hikari.com Brian J. McCartin, MYSTERIES OF THE EQUILATERAL TRIANGLE, First published 2010. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission of the publisher Hikari Ltd. ISBN 978-954-91999-5-6 Copyright c 2010 by Brian J. McCartin Typeset using LATEX. Mathematics Subject Classification: 00A08, 00A09, 00A69, 01A05, 01A70, 51M04, 97U40 Keywords: equilateral triangle, history of mathematics, mathematical bi- ography, recreational mathematics, mathematics competitions, applied math- ematics Published by Hikari Ltd Dedicated to our beloved Beta Katzenteufel for completing our equilateral triangle. Euclid and the Equilateral Triangle (Elements: Book I, Proposition 1) Preface v PREFACE Welcome to Mysteries of the Equilateral Triangle (MOTET), my collection of equilateral triangular arcana. While at first sight this might seem an id- iosyncratic choice of subject matter for such a detailed and elaborate study, a moment’s reflection reveals the worthiness of its selection. Human beings, “being as they be”, tend to take for granted some of their greatest discoveries (witness the wheel, fire, language, music,...). In Mathe- matics, the once flourishing topic of Triangle Geometry has turned fallow and fallen out of vogue (although Phil Davis offers us hope that it may be resusci- tated by The Computer [70]). A regrettable casualty of this general decline in prominence has been the Equilateral Triangle. Yet, the facts remain that Mathematics resides at the very core of human civilization, Geometry lies at the structural heart of Mathematics and the Equilateral Triangle provides one of the marble pillars of Geometry.