26 Mr. A. De Morgan on the Conic Octagram. [April 25, of Them, That 8 Is the Tangential of 4

Total Page:16

File Type:pdf, Size:1020Kb

26 Mr. A. De Morgan on the Conic Octagram. [April 25, of Them, That 8 Is the Tangential of 4 26 Mr. A. De Morgan on the Conic Octagram. [April 25, of them, that 8 is the tangential of 4. We have here the theorem that the third point of 1 and 7 is also the tangential of 4. Similarly, 10, 11, 13 are each of them (like 8) determined by two constructions ; 14, 16, 17, 19, each of them by three constructions, and so on; the number of constructions increasing by unity for each group of four numbers. And the theorem is, that these constructions, 2, 3 or more, as the case may be, give always one and the same point. Prof. Cayley mentioned that on a large figure of a cubic curve he had, in accordance with the theorem, constructed the series of points 1, 2, 4, 5, 7, 8, 10, 11, 13, 14. Prof. Sylvester made a communication " On the Theory of Re- siduals." Mr. Walker gave a proof of the Rectangle of Forces ; and demon- sti-afcion of tbe principle of the Parallelogram of Forces. These are new and interesting specimens of a class of proofs on the character of which mathematical opinion has been much divided. The interest of these proofs now turns on the question, how much and what there is of a physical character in the assumptions on which they are founded: in fact, what there is to prevent their being purely geometrical, and having a conclusion time of mathematical necessity. April Ihth, 18G7. Prof. SYLVESTER, President, in the Chair. M. Chasles was elected a Foreign Member of the Society. Prof. Clerk-Maxwell, F.R.S., Dr. Robertson, F.R.S.E., and Rev. Thos. Smith, M.A., were elected Members. The following paper was read :— On the Conic Octagram. By A. DE MORGAN, Esq., Vico-President. The most general theorem I require is the particular case I now state. If of the 1\? intersections of two n-ic curves n («—2) —^(w—3) (/i—4) lie on an («—2)ic, so also do j(n—8) (n—4) more of the inter- sections, which call adjuncts: and the remaining 2n intersections lie on a conic. When n = 3, and the cubics are ti"ilinear, we have Pascal's hexagram: we may retain this word as denoting, not a hexagon, but a method of drawing or completing a hexagon. Each value of n, if the «-ics be multilinear, gives a direct and inverse 2>i-gram : direct, when by an (yi — 2)ic taken at pleasure we deter- mine 2?i points of some conic of which nothing is given beforehand ; inverse, when, having given five determining points of a conic, wo 1867.] Mr. A. DG Morgan on the Oonic Octagram. 27 determine 2n—5 points more by an inverse construction which detcr- • mines a convenient (n—2)ic. When n = 3 or w = 4, there are no adjnnct intersections; and the inverse 2«-gram is the plainest inversion of a simple construction. But when « = 5 or upwards, the inverse 2?i-gram, though practicable, requires the introduction of an entirely new element. When n = 5, for example, and there is one adjuuet, the direct construction makes 14 intersections put the 15th on its proper place ; but, in the invei'se construction, the cubic cannot be discovered without finding it so as to satisfy the condition of the 15th intersection. Consequently, there are no other theorems of the family of Pascal's hexagram except only the octagram. I shall state the methods for: 1. the hexagram; 2. the double trigram, which is another form of the hexagram ; 3. the octagram; 4. the doable tetragram, which is not another form of the octagram ; 5. the direct form of the decagram. I signify lines by 1, 2, 3. &c.; points of intei-section by 12, 23, &c.; a line drawn through two points by 12 | 34, &c. And when A, B ai"e points on a line 3, A3B signifies as much of 3 as is between A and B ; and so on. 1. Hexagram. Direct: if the hexagon* 142536 have its alternate points (14, 25, 30) colliuear, then 123456 is in a conic; and the con- verse. Inverse: if 12345G be a conic hexagon, of which are known 61, 12, 23, 34, 45, and tho direction of 5, then (14 | 25 . 3) | 61 meets 5 in 56, the sixth point. After the data are laid down, two new lines determine one point. 2. Double trigram. This is a modification of the hexagi'am. which depends on Desargues's coaxal triangles. These triangles form a conic hexagon; that is, if 123, 456 be such triangles, 12, 23, 34, 45, 56, 61 aref co-conic. The inverse theorem is only worth mention for analogy : the construction is niore complex than the hexagram. Let 123 be an inscribed triangle, and 456 another, of which 45, 40, and the direction of 5, are given. From the intersection of 12 | 45 and 13 | 46 draw a line 7 to the intersection of 23 | 46 and 5. Then 72 | 13 meets 5 in the required point 56. The coaxal triangles are 45 (23 | 46) and 12 (72 | 13). And 7 is the Pascalian line of the hexagon 13, 40, 32, 21, 45, 56. 3. Octagram. Direct: choose a conic (usually a pair of straight * This presentation of the second hexagon was actually suggested to mo by observing that lsluisc Pascal lias two hrxagnuns, and the jocose inference that there ought to bu two hexagons in this theomn. My own names arB both oetagi'ams; but, though I bow ln-foro the coincidence, 1 have no suggestion to acknowledge. t Desargues was the teacher of Pascal, who expresses his obligation in terms of unusual strength. It is curtain, by tho date of printed satire, that tho theorem called la l'asai/c was known by that name bifore Pascal was twenty years old. I feel sure that Pascal's active genius was led by the coaxal triangles to tho hoxagram. 28 Mr. A. De Morgan on the Conic Octagram. [April 25, lines) as guide, and inscribe an octagon of which the successive chorda are numbered by procession by 5, rejecting 8; as, 16305274. Then 01, 12, 23, 34, 45, 56, 67, 70 are co-conic; or, 01234567 is an oc tagon inscribed in a conic. Inverse : in the conic octagon 01234567 are given 1, 2, 3, 4, and the directions of 0 and 5: required 56, 67, 70. Find 14, 50, 25, 30, 14 | 50, 25 | 30, 14 | 50 . 2, 14 | 50 . 3, 25 | 30 . 1, 25 | 30 . 4. Let 6 and 7 be 14 | 50 . 3 125 | 30 . 1 and 14 | 50 . 2 | 25 | 30 . 4. Then 56, 67, 70 will be on the conic, and 01234567 is an inscribed octagon. The guide is the system 14 | 50, 25 J 30 ; and three points are determined by four lines over and above data. By varying the directions of 0 and 5, and drawing the corresponding parts of all the constructions together, the laying down of a large number of points is very easy and rapid woi'k. 4. Double tetragram. When I explained the octagram to the Society, Mr. Cotterill suggested a known theorem which seemed likely <o be the direct octagram; but, on looking into it, I found it to be the direct double tetragram, which it suggested in another form. In a conic (usually two straight lines) inscribe two tetragrams 1630 and 5274; then 1234, 5670 are tetragons in another conic. Inversely: let the tetragon 1234 be given; and in 5670 the point 50 and the directions of 5 and 0. Find 51, 53, 02, 04, 51 | 02, 53 | 04; these lines are the guide. Let 6 and 7 be 53 | 04 . 2 | 51 | 02 . 4 and 53 | 04 . 11 51 | 02 . 3. Then 56, 67, 70 will bo on the conic, and 5670 is a second tetragon. Owing to our better command of language for a tetragon than for a hexagon, we may describe this last construction as follows:—Let one pair of opposite sides be bases, and the other pair sides; points in their lines being base-points and side-points. Given a tetragon, and a fifth point of the conic in which two lines meet, take the base-points in which the fifth line, and the side-points in which the sixth line cuts the tetragon. Join these two and two, base-point to side-point; in each of the joining lines there is a new base-point and a new side- point. Join these last, base-point to base-point, and side-point to Bide-point; the two lines so drawn complete the second tetragon. 5. Direct decagram. Take any three straight lines P, p, c; any curve of the third order would do as well. Capital letters are points on P ; small letters on p. Describe the decagon A0alB263C4c5D6d7E8e9A in the following way. Take A0a, olB, B26, fe3C, C4c at pleasure, and let the points in which 0, 1, 2, 3, 4, thus laid down, meet rs be 05,16, 27, 38,49. Then draw c5D through c and 05, meeting P in D, &c, as from the above description. It will at last be found that 9, as obtained from e and 49, passes through A.; or the adjunct falls as 1867.] Mr. Jenkins on an Arithmetical Theorem. 29 required by the theorem. This done, 03, 36, 69, 92, 25, 58, 81, 14, 47, 70, are co-conic; or 0369258147 is a conic decagon. Inci- dentally, we have solved the question of laying down fifteen points that shall lie five and five on three lines, and in two ways three and three on five lines.
Recommended publications
  • Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur Si Muove: Biomimetic Embedding of N-Tuple Helices in Spherical Polyhedra - /
    Alternative view of segmented documents via Kairos 23 October 2017 | Draft Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur si muove: Biomimetic embedding of N-tuple helices in spherical polyhedra - / - Introduction Symbolic stars vs Strategic pillars; Polyhedra vs Helices; Logic vs Comprehension? Dynamic bonding patterns in n-tuple helices engendering n-fold rotating symbols Embedding the triple helix in a spherical octahedron Embedding the quadruple helix in a spherical cube Embedding the quintuple helix in a spherical dodecahedron and a Pentagramma Mirificum Embedding six-fold, eight-fold and ten-fold helices in appropriately encircled polyhedra Embedding twelve-fold, eleven-fold, nine-fold and seven-fold helices in appropriately encircled polyhedra Neglected recognition of logical patterns -- especially of opposition Dynamic relationship between polyhedra engendered by circles -- variously implying forms of unity Symbol rotation as dynamic essential to engaging with value-inversion References Introduction The contrast to the geocentric model of the solar system was framed by the Italian mathematician, physicist and philosopher Galileo Galilei (1564-1642). His much-cited phrase, " And yet it moves" (E pur si muove or Eppur si muove) was allegedly pronounced in 1633 when he was forced to recant his claims that the Earth moves around the immovable Sun rather than the converse -- known as the Galileo affair. Such a shift in perspective might usefully inspire the recognition that the stasis attributed so widely to logos and other much-valued cultural and heraldic symbols obscures the manner in which they imply a fundamental cognitive dynamic. Cultural symbols fundamental to the identity of a group might then be understood as variously moving and transforming in ways which currently elude comprehension.
    [Show full text]
  • 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.
    [Show full text]
  • Compact Design of Modified Pentagon-Shaped Monopole Antenna for UWB Applications
    International Journal of Electrical and Electronic Engineering & Telecommunications Vol. 7, No. 2, April 2018 Compact Design of Modified Pentagon-shaped Monopole Antenna for UWB Applications Sanyog Rawat1, Ushaben Keshwala2, and Kanad Ray3 1 Manipal University Jaipur, India. 2 Amity University Uttar Pradesh, India. 3 Amity University Rajasthan, India. Email: [email protected]; [email protected]; [email protected] Abstract—In this work detailed analysis of a modified, introduction, and in Section II the initial antenna pentagon-shaped planar antenna is presented for ultra-wide geometry is presented and discussed. In the preceding bandwidth (UWB) applications. The proposed antenna was section, the modified pentagon-shaped antenna is designed on an FR-4 substrate and has a compact size of 12 elaborated and the results are discussed. × 22 × 1.6 mm3. It achieves an impedance bandwidth of 8.73 GHz (3.8–12.53 GHz) in the UWB range. The design has a II. ANTENNA GEOMETRY uniform gain and stable radiation pattern in the operating bandwidth. A. Initial Pentagram Star-shaped Antenna Index Terms—Golden angle, pentagram, pentagon, star- The antenna presented in this paper was designed on a shaped antenna FR-4 substrate, which was a partial ground on one side and a conducting patch on the opposite side. The antenna design is initially started with a pentagram star shape and I. INTRODUCTION then the shape was modified into a pentagon shape. The Nowadays, wireless communications and their high- pentagram is also called a pentangle, i.e., 5-pointed star speed data rate are becoming increasingly popular. The [13]. The geometry of the microstrip line fed star-shaped research in the field of microstrip antennas has also been monopole antenna is shown in Fig.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Fourth Lecture
    Fourth Knowledge Lecture of the New Hermetics The Symbol Of Venus and The Tree Of Life It embraces all ten Sephiroth on the Tree. It is a fitting emblem of the Isis of Nature. Since it contains all the Sephiroth its circle should be made larger then that of Mercury shown in a previous diagram. The Geomantic figures and their Zodiacal Attributions Planets, Colors, Lineal Figures Etc. No. Planet Color Shape Odor 3. Saturn Black Triangle Myrrh 4. Jupiter Blue Square Cedar 5. Mars Red Pentagram Pepper 6. Sun Yellow Hexagram Frankincense 7. Venus Green Septagram Benzoin 8. Mercury Orange Octagram Sandalwood 9. Moon Violet Enneagram Camphor Element Color Odor Fire Red Cinnamon Water Blue Cedar Air Yellow Sandalwood Earth Black Myrrh DIAGRAM 66 The Triangle The first linear shape, associated with the sephirah Binah and the planet Saturn. Medieval sorcerers used triangles to bind spirits because they believed the limiting force of the triangle would confine the spirit. The triangle may be used in any effort to restrict, structure or limit anything. The number three also indicates cycles and therefore time, also a limiting factor appropriate to Saturn. The Square The square is related to Chesed, whose number is four. A square implies the form of a castle or walled structure, society, prestige, rulership and other Jupiterean qualities. The square also symbolizes structure beyond walls, it represents a completion or perfection; a "square deal" and three "square meals" a day illustrate the archetypal idea verbally. Also the four elements working in balanced harmony. The Pentagram The pentagram is the Force of Mars, and the sephira Geburah.
    [Show full text]
  • Self-Dual Configurations and Regular Graphs
    SELF-DUAL CONFIGURATIONS AND REGULAR GRAPHS H. S. M. COXETER 1. Introduction. A configuration (mci ni) is a set of m points and n lines in a plane, with d of the points on each line and c of the lines through each point; thus cm = dn. Those permutations which pre­ serve incidences form a group, "the group of the configuration." If m — n, and consequently c = d, the group may include not only sym­ metries which permute the points among themselves but also reci­ procities which interchange points and lines in accordance with the principle of duality. The configuration is then "self-dual," and its symbol («<*, n<j) is conveniently abbreviated to na. We shall use the same symbol for the analogous concept of a configuration in three dimensions, consisting of n points lying by d's in n planes, d through each point. With any configuration we can associate a diagram called the Menger graph [13, p. 28],x in which the points are represented by dots or "nodes," two of which are joined by an arc or "branch" when­ ever the corresponding two points are on a line of the configuration. Unfortunately, however, it often happens that two different con­ figurations have the same Menger graph. The present address is concerned with another kind of diagram, which represents the con­ figuration uniquely. In this Levi graph [32, p. 5], we represent the points and lines (or planes) of the configuration by dots of two colors, say "red nodes" and "blue nodes," with the rule that two nodes differently colored are joined whenever the corresponding elements of the configuration are incident.
    [Show full text]
  • Capillary Self-Alignment of Polygonal Chips: a Generalization for the Shift-Restoring Force J
    Capillary self-alignment of polygonal chips: A generalization for the shift-restoring force J. Berthier,1 S. Mermoz,2 K. Brakke,3 L. Sanchez,4 C. Fr´etigny,5 and L. Di Cioccio4 1)Biotechnology Department, CEA-Leti, 17 avenue des Martyrs, 38054 Grenoble, France 2)STMicroelectronics, 850 rue Jean Monnet, 38926 Crolles Cedex, France 3)Mathematics Department, Susquehana University, 514 University avenue, Selingrove, PA 17870, USA 4)Microelectronic Department, CEA-Leti, 17 avenue des Martyrs, 38054 Grenoble, France 5)ESPCI, 10 rue Vauquelin, 75005, Paris, France (Dated: 6 May 2012) Capillary-driven self-alignment using droplets is currently extensively investigated for self-assembly and mi- croassembly technology. In this technique, surface tension forces associated to capillary pinning create restor- ing forces and torques that tend to bring the moving part into alignment. So far, most studies have addressed the problem of square chip alignment on a dedicated patch of a wafer, aiming to achieve 3D microelectronics. In this work, we investigate the shift-restoring forces for more complex moving parts such as regular { convex and non-convex { polygons and regular polygons with regular polygonal cavities. A closed-form approximate expression is derived for each of these polygonal geometries; this expression agrees with the numerical results obtained with the Surface Evolver software. For small shifts, it is found that the restoring force does not depend on the shift direction or on the polygonal shape. In order to tackle the problem of microsystem pack- aging, an extension of the theory is done for polygonal shapes pierced with connection vias (channels) and a closed form of the shift-restoring force is derived for these geometries and again checked against the numerical model.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Complementary N-Gon Waves and Shuffled Samples Noise
    Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx2020),(DAFx-20), Vienna, Vienna, Austria, Austria, September September 8–12, 2020-21 2020 COMPLEMENTARY N-GON WAVES AND SHUFFLED SAMPLES NOISE Dominik Chapman ABSTRACT A characteristic of an n-gon wave is that it retains angles of This paper introduces complementary n-gon waves and the shuf- the regular polygon or star polygon of which the waveform was fled samples noise effect. N-gon waves retain angles of the regular derived from. As such, some parts of the polygon can be recog- polygons and star polygons of which they are derived from in the nised when the waveform is visualised, as can be seen from fig- waveform itself. N-gon waves are researched by the author since ure 1. This characteristic distinguishes n-gon waves from other 2000 and were introduced to the public at ICMC|SMC in 2014. Complementary n-gon waves consist of an n-gon wave and a com- plementary angular wave. The complementary angular wave in- troduced in this paper complements an n-gon wave so that the two waveforms can be used to reconstruct the polygon of which the Figure 1: An n-gon wave is derived from a pentagon. The internal waveforms were derived from. If it is derived from a star polygon, angle of 3π=5 rad of the pentagon is retained in the shape of the it is not an n-gon wave and has its own characteristics. Investiga- wave and in the visualisation of the resulting pentagon wave on a tions into how geometry, audio, visual and perception are related cathode-ray oscilloscope.
    [Show full text]