Magic Squares

In this work the magic squares of order n are addressed from the perspective of linear algebra, vector space, "Magic Space" and some of its basis is presented.

The case of sparse systems of equations that having more than one qualitatively different row echelon form.

This paper brings numbers in such a way that both sides of the expressions are with same digits. One side is numbers with power, while other side just with numbers, such as, a^b+c^d +.. =ab +cd+..., etc. The the expressions studies are with positive and negative signs. Work is done for two to five terms expressions. Due to high quantity of numbers, in some cases, powers and bases both are considered bigger than one.

A magic square is an n × n matrix of the integers from 1 to n2 such that the sum of every row, column, and diagonal is the same. There are several ways for generating certain types of magic squares, which results different magic squares of the same order. But until the work done by Abiyev [A.A. Abiyev, Sayılı sihirli karelerin doğal şifresi (in Turkish), Enderun Ofset Matbaacılık, ISBN: 975-95318-3-6, Ankara, 1996] there was not a known general methodology for generating magic squares of any order. Moreover, Abiyev's method is not restricted to a certain type of numbers (real, integer, etc.). By his method magic squares of any order can be obtained for any type of numbers including complex numbers. Abiyev's work is now its maturing stage and magic squares generated by his method shows some very interesting symmetrical properties, which are not possible to obtain via other techniques. These properties may open some research directions for several scientific disciplines like genetic engineering, operational research, intelligent computation, physics, etc. Such an attempt is made in this paper to show how center of mass can be analysed by using Abiyev's magic squares.

Quali sono le aspettative ed opinioni degli studenti di cultura cinese sulla scuola e sulla matematica? Quali le concezioni di scuola e di studio che ereditano delle famiglie? In questo volume si tenta di offrire risposte a questi interrogativi per facilitare l'azione didattica degli insegnanti alle prese con classi sempre più multiculturali. Vi compare inoltre una lunga rassegna di temi, testi e personaggi della tradizione matematica cinese ed uno studio degli algoritmi specifici di questa cultura. Introduzione di Filippo Spagnolo (GRIM Palermo).

This paper will conduct a detail analyze on Freemasonry's symbolism codified to The Great Seal of the United States obverse and reverse sides, The Apotheosis of George Washington, The Centennial Great Seal of 1882, The Capitol Building and the mapping out of Washington DC.

Here I analyze Freemasonry ongoing enterprise of codifying esoteric symbols into the streets and monuments of Washington DC.  Most of the symbols I discuss have never been videoed or written about by previous authors or film makers.

Taking the view that all true spirituality is inherently Anarchist at it's core, this pdf book is a comprehensive examination of the many links between Anarchism, the Occult and Mysticism. It is presented within the broad framework of a modern myth and poetry and tells the story of a young punk Anarchist writer, who discovers the magickal nature of reality and so decides to write a book to spread awareness and spiritual anarchism. However it also heavily researched and like so many spiritualities, it uses this mythology as a vehicle for transmitting the mysteries, with the majority of the book being the prosaic teachings of the protagonist. It assumes no knowledge on the part of the reader and provides introductions to the topics discussed, before going on to more original and in depth ideas. While a history of spiritual anarchists and the theocratic black magicians that oppose them is given to provide context, this book focuses more on integrating mystical and magickal teachings from around the world and creating a detailed and modern Anarchist spirituality that suits the individual. It also provides a variety of practical ritual meditative techniques for creating an Anarchist consciousness, as well as using magick for activist and Anarchist purposes. If you were ever looking for an Anarchist grimoire, this is it! One of the main themes of the book is the global use of psychedelics or “entheogens” for spiritual purposes and how they can be utilized for overcoming the counter-revolutionary aspects of the mind. The book also explains how Occult techniques can be used to help direct the trips for expanding consciousness and performing magick. (especially DMT and Salvia Divinorum) Another primary theme is how language and “memes” apply to fractals, how Noam Chomsky's notion of “universal grammar” was actually discovered by Occultists along with the scientific method, but also how language shapes human consciousness, politics and the world around us. Other topics include an Occult interpretation of the historical origins of the monetary and legal systems, as well as the origins of Judaism, Christianity and governments through theocracy. Inside you will find explanations for how art and “memes” can be used for Anarchist propaganda, how ritual and magick apply to “neuroplasticity” and how all these things can literally shape our brains either to create free will, or as a form of brainwashing. This book explains theocracy as the root of all hierarchy which is itself the result of a kind of Obsessive compulsive memetic brain virus created by "black magicians." All this and much more is found within, while the epilogue explains groundbreaking political interpretations of “The Cube of Space,” Solomonic magick, Enoch and the Watchers, Gnosticism and world mythology in general. Here the author shows how this mythology relates to a meteorite death cult and ancient astronomical events, atomic theory, numerology and sacred geometry. It also gives advanced ideas on reconstructing an old world form of Alchemical Tantric Shamanism and how magick squares can be utilized with set theory to create the philosopher's stone in your brain!
A chapter by chapter synopsis is given in the table of contents

This essay shows all the most important numbers associated with the 7 magick squares of Cornelius Agrippa, along with others, can be found on an extended ordinal counting sequence version of the Tetractys of Pythagoras. That is by counting out numbers in a triangular form the "triangular numbers" are highlighted in the pattern and this also shows the numbers of magick squares. A detailed look at this numerical Tetractys is given, with interpretations on what it could mean for Pythagoras, the Kabbalistic cube of space, along with cubic numbers and even the Sri Yantra and Chakra systems. (Showing Kabbalistic numbers in the Sri Yantra) Building upon the work of David Fideler and his book "Jesus Christ Sun of God" gematria is used to interpret the biblical parables of 153 fish and the feeding of 5000.  All this shows how magick squares and their associated numerology is the basis for various traditions of number mysticism.  While there is far more to explore in terms of interpreting texts, with the evidence provided, the main thesis shows how the magick squares undeniably fit with the Tetractys and that Polygonal numbers in general are also very important.

In the earliest concrete examples of magic squares from Chinese, Indian, and Arabic literature, the squares are valued not simply as mathematical curiosities but as objects of wondrous powers. Recent research on early Arabic awfāq (magic squares) literature has identified a large corpus with several known authors, but studies of this corpus tend to focus on mathematical analyses. This article sketches a cultural history of awfāq, which contextualizes them within wider Islamicate culture, situating awfāq literature within the oeuvres of the scholars who produced it, and locating those scholars within networks of patrons, colleagues, and institutions. Beginning with magic squares in pre-Islamic cultures and the continuation of those traditions, this article explores Islamicate legends of the origins and transmission of the awfāq and surveys the corpus of Arabic awfāq literature from the mid-third/ninth to the late sixth/twelfth centuries in an attempt to grasp the background, context, and significance of the great popularity of awfāq in talismanic practices associated with the Science of Letters (ʿilm al-ḥurūf).

This essay shows all the most important numbers associated with the 7 magick squares of Cornelius Agrippa, along with others, can be found on an extended ordinal counting sequence version of the Tetractys of Pythagoras. That is by counting out numbers in a triangular form the "triangular numbers" are highlighted in the pattern and this also shows the numbers of magick squares. A detailed look at this numerical Tetractys is given, with interpretations on what it could mean for Pythagoras, the Kabbalistic cube of space, along with cubic numbers and even the Sri Yantra and Chakra systems. (Showing Kabbalistic numbers in the Sri Yantra) Building upon the work of David Fideler and his book "Jesus Christ Sun of God" gematria is used to interpret the biblical parables of 153 fish and the feeding of 5000. All this shows how magick squares and their associated numerology is the basis for various traditions of number mysticism. While there is far more to explore in terms of interpreting texts, with the evidence provided, the main thesis shows how the magick squares undeniably fit with the Tetractys and that Polygonal numbers in general are also very important.

Si tratta dell'analisi di un crittogramma dalla probabile ascendenza cristiana o giudaico-cristiana, che viene descritto come “Quadrato SATOR” o “Quadrato ROTAS”, o ancora “Quadrato magico pompeiano”, in quanto le sue più antiche raffigurazioni sono state rinvenute appunto nel corso degli scavi della cittadina distrutta dall'eruzione del Vesuvio del 79 d.C.

The Beinecke Rare Books Reading Room at Yale University houses an important collection of Arabic papyri and paper documents. Among them are at least two printed texts: RCtYBR inv. 2016 and RCtYBR inv. 2367. The latter is a small fragment containing but a few words, and is therefore of little value, while the former is a large fragment of an amulet, and is of considerable interest because it contains two magic squares, that is, squares in which numbers are arranged in such a way as to produce a constant sum in all rows and columns (Ar. wafq, pi. awfaq). Although the use of magic squares in amulets is well attested through theoretical works such as al-Buni's Shams al-ma(arif al-kubra, and through manuscript specimens, only one other example of a block-printed square is known. What is more, the present piece contains a rare occurrence of a magic square of the order 13 - perhaps the only such occurrence in all amulets published to date.

Este documento demuestra el descubrimiento de patrones ocultos en los cuadrados mágicos, estos forman figuras geométricas sorprendentes Cuando reducimos los valores de un cuadrado mágico a los 9 principales dígitos, este forma 3 patrones. Dos opuestos complementarios, (estos se necesitan entre sí para lograr la armonía) y un tercero llamado equitativo el cual está organizado y en equilibrado en sí mismo.

Quaderno n. 1 –– Conferenze Primaverili 1995, stampato per conto del Circolo da De Bastiani ed., Vittorio Veneto, settembre 1995 –– Sommario: (1) IMPERIO Loredana, Federico II, tra Mito e Storia pp. 3-20; (2) GUSSO Massimo, Il Quadrato “Magico” pompeiano pp. 21-35; (3) CADEDDU Lorenzo,1945-1975. Da Trieste a Osimo pp. 36-59; (4) RUZZA Vincenzo, Eretici e scismatici, tra Piave e Livenza. In età medioevale e moderna pp. 60-86; (5) POSOCCO Franco, Cartografia e Territorio. Il contributo della documentazione aerofotogrammetrica nella storia del paesaggio veneto pp. 87-96

楊輝（約1238-1298）為南宋末年的數學家，其著作《續古摘奇算法》（1275）中以「縱橫圖」一詞，稱呼吾人今日所知的幻方，自此之後，縱橫圖開始出現在中國歷代的算書中。數學史家Schuyler Cammann（1912-1991）認為楊輝乃是受到阿拉伯幻方的影響，但當真如此嗎？是故，本文之主旨乃是探究楊輝 之縱橫圖是否隱含著阿拉伯數學的影響。 縱橫圖是否隱含著阿拉伯數學的影響。 縱橫圖是否隱含著阿拉伯數學的影響。 縱橫圖是否隱含著阿拉伯數學的影響。 縱橫圖是否隱含著阿拉伯數學的影響。 縱橫圖是否隱含著阿拉伯數學的影響。 縱橫圖是否隱含著阿拉伯數學的影響。 筆者將藉由對楊輝的時代背景、幻方的構造方式及著作中隱含的宇宙觀思想，與同時代的阿拉伯幻方進行比對及分析，期以得出可能的解答。

Resumen: Este documento demuestra el descubrimiento de patrones ocultos en los cuadrados mágicos, estos forman figuras geométricas sorprendentes Cuando reducimos los valores de un cuadrado mágico a los 9 principales dígitos, este forma 3 patrones. Dos opuestos complementarios, (estos se necesitan entre sí para lograr la armonía) y un tercero llamado equitativo el cual está organizado y en equilibrado en sí mismo.

Esotericism is an extremely oblique opaque hidden science scribed into the texts of the Torah (Old and New Testaments). Most people believe that the Torah represents the five books of Moses; however, Judaism declares that any commentary on the Torah is the Torah. In the very beginning of the Prologue of the ZOHAR i the first topic is THE LILY and this is the most clarified commentary on the Torah's text that I have ever come across; because, the ZOHAR'S commentary (THE LILY) literally outlines the Lily mathematically in the first two verses of Genesis and the first two words in the first day of creation. 'Between the first and second mentions of the word ELOHYM are thirteen (13) words, which symbolizes thirteen (13) petals of a Lily. And between the second and third mention of the word ELOHYM are five (5) words that symbolizes five (5) leaves that surrounds the Lily. This Lily is the cup of benediction that must rest on five (5) fingers.' I have inserted the images of the SHECHINAH ‫,שכינה‬ which has the gematria value of three-hundred and eighty-five (385). The ten magic squares also has the gematria value of three-hundred and eighty-five (385).

In this paper, we present an identity which generalizes Vajda's and D'ocagne's identities involving Fibonacci sequence. Vajda's and D'ocagne's identities are known to contain 3 and 2 variables respectively, but the identity to be presented contains 7 variables. Binet's formula for generating the nth term of Fibonacci sequence will be used in proving the identity.

In line with the international efforts for communication with extraterrestrial civilizations, this article shows how to use the architectonic magic codes in order to communicate with the extraterrestrials that might have different alphabetical domains. It shows an example on how to convert an architectonic magic code into a musical piece using the relativistic musical language of the cosmic wide web from the pitch of La# major.

Representations of SO (4, 2) are constructed using 4×4 and 2×2 matrices with elements in ℍ' ⊗ ℂ and the known isomorphism between the conformal group and SO (4, 2) is written explicitly in terms of the 4×4 representation. The Clifford algebra structure of SO (4, 2) is briefly discussed in this language, as is its relationship to other groups of physical interest.

from: «Rewriting Recent Music History: the Development of Early Serialism. 1947–1957», ed. by Mark Delaere, Peeters, Leuven-Walpole, MA, 2011.
http://www.peeters-leuven.be/search_serie_book.asp?nr=246

Largely based on the study of extant sketches, the article aims at unveiling the compositional technique developed by Bruno Maderna in the context of the generalized renewal of the musical language in post-WWII Europe. Well trained in a solid compositional discipline, Maderna adopts the 12-tone technique while reinventing ideas and procedures derived from ancient examples (e.g. Renaissance polyphony). In the very early 1950s he finds his own way to serialism even managing to integrate methods derived both from Schoenberg and Hindemith.

Order in psychosocial space -- and strategic design implications
Values: Cannon-balls? Wealth symbols? Ball-games? Strange attractors?
Use of distinctive geometries as mapping surfaces for value-goals
Polyhedral mapping reconciling value-goals and their antitheses in the light of ball-games
Global mapping of 17 value-goals alone
Global mapping of 17 value-goals interacting with their 17 problematic opponents
Unusually memorable mapping possibilities of a tetrahedral geodesic spheres
Interplay of value-goals -- integrating opposing "hemispheres of the global brain"
Global mapping in 4D of interactions of competing value-goals?
Integrative comprehension of disparate global value-goals through sphere-packing
Requisite 12-fold organization implied by the 17-fold articulation of value goals -- as its "stellations"?
Unexplored cognitive implications of 17 Sustainable Development Goals -- through "magic squares"?

An magic square is N*N matrix containing integers and addition result of each row, column and diagonally get the same value. The commonly used methods of constructing magic squares are the cross diagonals method for doubly even. But these method cannot be used to find higher order magic square. In this article we will find the method to solve higher order doubly even magic squares.

Para Siglind Bruhn, o conceito de ekphrasis, usado nas artes plásticas e na literatura, pode ser aplicado também a certas obras musicais. Partindo dessa concepção, desenvolvida pela autora na obra "Musical Ekphrasis: composeres responding to poetry and painting", este artigo investiga a apropriação que o compositor cubano Leo Brouwer (1939 - ) faz, no primeiro movimento de sua Sonata para violão solo, de certos procedimentos composicionais utilizados pelo pintor suíço Paul Klee (1879–1940) em sua série de pinturas intitulada Quadrados Mágicos, caracterizando, assim, uma ekphrasis.

Abstract. For Siglind Bruhn, the concept of ekphrasis, used on visual arts and literature, can be applied to certain pieces of music. Starting from this concept, developed by Bruhn on the work "Musical Ekphrasis: composeres responding to poetry and painting", the present article investigates the appropriation that the Cuban composer Leo Brouwer (1939 - ) makes, on the first movement of his Sonata for solo guitar, of certain compositional procedures used by the Swiss painter Paul Klee (1879–1940) in his series of pictures entitled Magic Squares, characterizing, thus, an ekphrasis. (article only in Portuguese)

Para Siglind Bruhn, o conceito de ekphrasis, usado nas artes plásticas e na literatura, pode ser aplicado também para certas obras musicais. Partindo dessa concepção, desenvolvida pela autora na obra Musical Ekphrasis: composeres responding to poetry and painting este artigo investiga a apropriação que o compositor cubano Leo Brouwer (1939 - ) faz, no primeiro movimento de sua Sonata para violão solo, de certos procedimentos composicionais utilizados pelo pintor suíço Paul Klee (1879-1940) em sua série de pinturas intitulada Quadrados Mágicos, caracterizando, assim, uma ekphrasis.

Butterfly dream: "inside-outside"? Psychosocial implications of indeterminacy: wave versus particle? Cognitive shapeshifting? Betwixt and between: the liminal art? Being a particle versus Being a wave-and the alternation between Experience of "flying" versus "being stoned" Flow psychology and being "in the zone" From "in-the-box" to "out-of-the-box" Turtle of Lo Shu embodying traditional magic squares Creating one's own reality through aesthetics Challenge for both turtle and frog in encountering black swans? Insights from a carp ensuring complementarity of radically distinct perspectives? Human evolution: Homo conjugens and Homo undulans?

In this communication the square microstrip antenna consisting of dual open stubs is presented for quad band operation. The antenna operates between 2.88 to 8.55 GHz. The antenna is constructed with its structure of dimension 8 X 5 X 0.16 cm3 . The microstripline feed arrangement along with quarter wave transformer is used to excite the antenna. The antenna exhibits a broadside and linear radiation characteristics. The peak gain of 3.21 dB is obtained in the operating band. The results are presented and discussed. This antenna may find its applications in WLAN and Wimax communication system.