There is little opportunity in classrooms today for teachers to explore the amazing applications of mathematics outside the curriculum. This book is intended to show how mathematics manifests itself in areas that most people are unaware of. One can even revel in the history of how our number system evolved and how that has enabled us to define the beauty in mathematics as well as in art, architecture, nature, and beyond.

The first two chapters of this book introduce the Fibonacci numbers and investigate their amazing relationships and applications in our general environment. The following four chapters focus on the Golden Ratio and the Golden Rectangle, exploring how they manifest all around us, often hiding in plain sight: in everything from architectural wonders such as the Taj Mahal to coin design, and from Greek vases to petal formation. We conclude our enjoyable journey through these mathematical wonders by considering conic sections and how they explain many aspects of everyday life, such as radar dishes, headlight reflectors, and whispering halls. This exposure to aspects of mathematics that are usually bypassed in the school curriculum will provide high school students, teachers, and general readers with an opportunity to truly appreciate the power and beauty of mathematics.

Alfred S. Posamentier is currently Distinguished Lecturer at New York City College of Technology of the City University of New York. Previously, he was the Executive Director for Internationalization and Sponsored Programs at Long Island University, New York. This was preceded by a five-year period, where he was Dean of the School of Education and tenured Professor of Mathematics Education at Mercy University, New York. He is now also Professor Emeritus of Mathematics Education at The City College of the City University of New York, and former Dean of the School of Education, where he was tenured for 40 years. He is the author and co-author of more than 80 mathematics books for teachers, secondary and elementary school students, and the general readership. Dr. Posamentier is also a frequent commentator in newspapers and journals on topics relating to education.
After completing his B.A. degree in mathematics at Hunter College of the City University of New York in 1964, he took a position as a teacher of mathematics at Theodore Roosevelt High School (Bronx, NY), where he focused his attention on improving the students’ problem-solving skills and at the same time enriching their instruction far beyond what the traditional textbooks offered. During his six-year tenure there, he also developed the school’s first mathematics teams (both at the junior and senior level). He is still involved in working with mathematics teachers and supervisors, nationally and internationally, to help them maximize their effectiveness.
During this time, he earned an M.A. degree at the City College of the City University of NewYork in 1966. Immediately upon joining the faculty of the City College in 1970, he began to develop in-service courses for secondary school mathematics teachers, including such special areas as recreational mathematics and problem-solving in mathematics. As Dean of the City College School of Education for 10 years, his scope of interest in educational issues covered the full gamut. During his tenure as dean, he took the School of Education from the bottom of the New York State rankings to the top with a perfect NCATE accreditation assessment in 2009. He achieved the same success in 2014 at Mercy University, which received both NCATE and CAEP accreditation during his leadership as Dean of the School of Education.
In 1973, Dr. Posamentier received his Ph.D. from Fordham University (New York) inmathematics education and has since extended his reputation in mathematics education to Europe. He has been visiting professor at several European universities in Austria, England, Germany, Czech Republic, and Poland, while at the University of Vienna he was Fulbright Professor (1990). In 1989 he was awarded an Honorary Fellow at the South Bank University (London, England). In recognition of his outstanding teaching, the City College Alumni Association named him Educator of the Year in 1994, and in 2009. New York City had the day, May 1, 1994, named in his honor by the President of the New York City Council. In 1994, he was also awarded the Grand Medal of Honor from the Republic of Austria, and in 1999, upon approval of Parliament, the President of the Republic of Austria awarded him the title of University Professor of Austria. In 2003 he was awarded the title of Ehrenbürger (Honorary Fellow) of the Vienna University of Technology, and in 2004 was awarded the Austrian Cross of Honor for Arts and Science, First Class from the President of the Republic of Austria. In 2005 he was inducted into the Hunter College Alumni Hall of Fame, and in 2006 he was awarded the prestigious Townsend Harris Medal by the City College Alumni Association. He was inducted into the New York StateMathematicsEducator’sHall of Fame in 2009, in 2010 he was awarded the coveted Christian-Peter-Beuth Prize in Berlin, and in 2017 he received the Summa Cum Laude nemine discrepante Award from Fundacion Sebastian, A.C. in Mexico City. He has taken on numerous important leadership positions in mathematics education locally. He was a member of the New York State Education Commissioner’s Blue-Ribbon Panel on the Math- A Regents Exams in 2003, and in 2014–15 he was on the Commissioner’s Mathematics Standards Committee, which redefined the Standards for New York State, and he also served on the New York City schools’ Chancellor’s Math Advisory Panel. Dr. Posamentier is a leading commentator on educational issues and continues his long time passion of seeking ways to make mathematics interesting to both teachers, students and the general public.


Günter J.Maresch is currently Professor ofMathematics Education and Vice-Dean of the Faculty of Digital and Analytical Sciences at the University of Salzburg (Austria). His research topics are spatial ability, spatial thinking training and diagnosis, descriptive geometry, computer-aided design (CAD), new digital media, didactical principles, and curriculum development. Dr. Maresch studied descriptive geometry andmathematics at the Vienna University ofTechnology (Austria) and after completing hisMaster’s degree (1994), he took a position as a teacher ofmathematics, descriptive geometry, and information and communication technology (ICT) at secondary schools in the province of Salzburg (Austria). Dr.Maresch was a lecturer at the University of Salzburg from 1998 to 2014, where he taught courses on spatial ability, descriptive geometry, computer-aided design (CAD), new digital media, and curriculum development.
Dr.Maresch completed his Ph.D. studies at the University of Salzburg (Austria) in 2004. His Ph.D. thesis focused on e-learning and computer-aided design as well as the development of a didactical concept for increasing the quality of geometry instruction. He continues to provide professional programs in many countries, including Austria, Germany, Finland, Netherlands, Denmark, Spain, Hungary, and South Africa. From 2014 to 2016, he was an Assistant Professor for Mathematics Education and Geometry at the University of Salzburg. In 2016,
Dr. Maresch was awarded the rank of Associate professor for Mathematics Education at the University of Salzburg.
Since 1998 Dr. Maresch has provided lecturers at more than 300 conferences and teacher in-service training programs in the form of keynotes, lectures, workshops, and seminars,which were focused on geometry (spatial ability, curriculum, new media, competence model(s), CAD trainings, etc.), mathematics (educational standards, competence model, didactical principles, etc.), ICT (new digital media, learning platforms, computeraided design, etc.), principal conferences and school inspector conferences (sustainability of teacher in-service trainings, etc.), and supervising teacher internship (centralized final exams, prescientific research paper, pedagogical competences, etc.). Dr. Maresch completed studies to earn a teaching certificate for ICT at the local school board of Salzburg (1998). He further qualified for the IntelMaster Teacher Certificate (2001) and completed the Leadership Academy Program (2008–2009) offered by the Austrian Ministry of Education. From 1998 to 2007, Dr. Maresch was the Teachers’ Mentor for teachers of descriptive geometry, mathematics, and ICT in the province of Salzburg. He also organized more than 50 conferences and teacher in-service training programs during this time.
In 2012, Dr.Maresch was named “Educator of theYear” at theworldwide competition of Bentley Systems, and in 2020, he was one of the winners of the “Digital Teaching Award” of the University of Salzburg. From 2007 to 2012, Dr. Maresch was a member of the local education management board and the head of the institute for lifelong learning for upper secondary schools at the Salzburg University College of Teacher Education and during this time he organized more than 700 teacher in-service training programs for teachers and headmasters ofAustrian lower and upper secondary schools per year. Since 2002, he has been a member of the curriculum board of the Austrian Ministry of Education—responsible for the development of the curricula for the secondary schools. In summary, he has published more than 100 research papers and 20 books for students, researchers and teachers of mathematics, and books on geometry for the general readership.
Dr. Maresch is the inventor and chief developer of the international online spatial thinking training and diagnosis platform RIF (https://rif4you.eu/en). More than 3 million tasks have been completed on the free platform by students from 49 countries around the world.

Preface
About the Authors
Acknowledgements
Introduction to the Fibonacci Numbers
Fibonacci Numbers in Everyday Occurrences
Introduction to the Golden Ratio
The Aesthetics of the Golden Rectangle in Architecture
The Aesthetics of the Golden Rectangle in Everyday Objects
Geometry in Nature
Conic Sections
Index