1. Menz HB, Lord SR, Fitzpatrick RC. Acceleration patterns of the head and pelvis when walking on level and irregular surfaces. Gait and Posture. 2003;18(1):35–46.[PubMed]
2. Bruijn SM, Meyns P, Jonkers I, Kaat D, Duysens J. Control of angular momentum during walking in children with cerebral palsy. Research in Developmental Disabilities. 2011;32(6):2860–2866.[PubMed]
3. Iosa M, Marro T, Paolucci S, Morelli D. Stability and harmony of gait in children with cerebral palsy. Research in Developmental Disabilities. 2012;33(1):129–135.[PubMed]
4. Borghese NA, Bianchi L, Lacquaniti F. Kinematic deferminants of human locomotion. Journal of Physiology. 1996;494(3):863–879.[PMC free article][PubMed]
5. Reisman DS, Block HJ, Bastian AJ. Interlimb coordination during locomotion: what can be adapted and stored? Journal of Neurophysiology. 2005;94(4):2403–2415.[PubMed]
6. Cappozzo A. Low frequency self-generated vibration during ambulation in normal men. Journal of Biomechanics. 1982;15(8):599–609.[PubMed]
7. Iosa M, Mazzà C, Frusciante R, et al. Mobility assessment of patients with facioscapulohumeral dystrophy. Clinical Biomechanics. 2007;22(10):1074–1082.[PubMed]
8. Giakas G, Baltzopoulos V. Time and frequency domain analysis of ground reaction forces during walking: an investigation of variability and symmetry. Gait and Posture. 1997;5(3):189–197.
9. Lamoth CJC, Meijer OG, Wuisman PIJM, van Dieën JH, Levin MF, Beek PJ. Pelvis-thorax coordination in the transverse plane during walking in persons with nonspecific low back pain. Spine. 2002;27(4):E92–E99.[PubMed]
10. Perry J. Gait Analysis: Normal and Pathological Function. Thorofare, NJ, USA: Slack Incorporated; 1992.
11. Kirtley C. Clinical Gait Analysis; Theory and Practice. Philadelphia, Pa, USA: Elsevier; 2006.
12. Winter DA, Patla AE, Frank JS, Walt SE. Biomechanical walking pattern changes in the fit and healthy elderly. Physical Therapy. 1990;70(6):340–347.[PubMed]
13. Shemmell J, Johansson J, Portra V, Gottlieb GL, Thomas JS, Corcos DM. Control of interjoint coordination during the swing phase of normal gait at different speeds. Journal of NeuroEngineering and Rehabilitation. 2007;27(4, article 10)[PMC free article][PubMed]
14. Riener R, Rabuffetti M, Frigo C. Stair ascent and descent at different inclinations. Gait and Posture. 2002;15(1):32–44.[PubMed]
15. Wright RB, Yoder DM, Costa JL, Andriacchi TP. Characterization of gait parameters in adult-onset myotonic dystrophy: abnormal hip motion. Archives of Physical Medicine and Rehabilitation. 1995;76(1):33–38.[PubMed]
16. Abernethy B, Hanrahan SJ, Kippers V, Mackinnon LT, Pandy MG. The Biophysical Foundations of Human Movement. 2nd edition. Melbourne, Australia: Palgrave Macmillan; 2005.
17. Lythgo N, Wilson C, Galea M. Basic gait and symmetry measures for primary school-aged children and young adults. II: walking at slow, free and fast speed. Gait and Posture. 2011;33(1):29–35.[PubMed]
18. Grattan Guinnes I. Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. The Johns Hopkins University Press; 2003.
19. Hemenway P. Divine Proportion: Phi in Art, Nature, and Science. New York, NY, USA: Sterling Publishing; 2005.
20. Pacioli L. De Divina Proportione. Venice, Italy: Paganinus de Paganinus; 1509.
21. Okabe T. Physical phenomenology of phyllotaxis. Journal of Theoretical Biology. 2011;280(1):63–75.[PubMed]
22. Gosling E. Bivalve Molluscs. Biology, Ecology and Culture. Oxford, UK: Blackwell Publishing; 2003.
23. Coldea R, Tennant DA, Wheeler EM, et al. Quantum criticality in an ising chain: experimental evidence for emergent e8 symmetry. Science. 2010;327(5962):177–180.[PubMed]
24. Yamagishi MEB, Shimabukuro AI. Nucleotide frequencies in human genome and Fibonacci numbers. Bulletin of Mathematical Biology. 2008;70(3):643–653.[PubMed]
25. Staff L, Hurd P, Reale L, Seoighe C, Rockwood A, et al. The hidden geometries of the Arabidopsis thaliana epidermis. Plos ONE. 2012;7e43546 [PMC free article][PubMed]
26. Woldenberg MJ, O’Neill MP, Quackenbush LJ, Pentney RJ. Models for growth, decline and regrowth of the dendrites of rat Purkinje cells induced from magnitude and link-length analysis. Journal of Theoretical Biology. 1993;162(4):403–429.[PubMed]
27. Ferring V, Pancherz H. Divine proportions in the growing face. American Journal of Orthodontics and Dentofacial Orthopedics. 2008;134(4):472–479.[PubMed]
28. Ricketts RM. Divine proportion in facial esthetics. Clinics in Plastic Surgery. 1982;9(4):401–422.[PubMed]
29. Russell PA. The aesthetics of rectangle proportion: effects of judgment scale and context. American Journal of Psychology. 2000;113(1):27–42.[PubMed]
30. Davis RB, III, Õunpuu S, Tyburski D, Gage JR. A gait analysis data collection and reduction technique. Human Movement Science. 1991;10(5):575–587.
31. Ounpuu S. The biomechanics of walking and running. Clinics in Sports Medicine. 1994;13(4):843–863.[PubMed]
32. Peppe A, Chiavalon C, Pasqualetti P, Crovato D, Caltagirone C. Does gait analysis quantify motor rehabilitation efficacy in Parkinson’s disease patients? Gait and Posture. 2007;26(3):452–462.[PubMed]
33. Reynolds NC, Jr., Myklebust JB, Prieto TE, Myklebust BM. Analysis of gait abnormalities in Huntington disease. Archives of Physical Medicine and Rehabilitation. 1999;80(1):59–65.[PubMed]
34. Kuan T, Tsou J, Su F. Hemiplegic gait of stroke patients: the effect of using a cane. Archives of Physical Medicine and Rehabilitation. 1999;80(7):777–784.[PubMed]
35. Wang X, Wang Y. Gait analysis of children with spastic hemiplegic cerebral palsy. Neural Regeneration Research Journal. 2012;7:1578–1584.[PMC free article][PubMed]
36. Ivanenko YP, Cappellini G, Dominici N, Poppele RE, Lacquaniti F. Modular control of limb movements during human locomotion. Journal of Neuroscience. 2007;27(41):11149–11161.[PubMed]
37. Zehr EP. Neural control of rhythmic human movement: the common core hypothesis. Exercise and Sport Sciences Reviews. 2005;33(1):54–60.[PubMed]
38. Ivanenko YP, Poppele RE, Lacquaniti F. Distributed neural networks for controlling human locomotion. Lessons from normal and SCI subjects. Brain Research Bulletin. 2009;78(1):13–21.[PubMed]
39. Graham Brown TG. The intrinsic factors in the act of progression in the mammal. Proceedings of the Royal Society B. 1911;84:308–319.
40. McCrea DA, Rybak IA. Organization of mammalian locomotor rhythm and pattern generation. Brain Research Reviews. 2008;57(1):134–146.[PMC free article][PubMed]
41. Orlovsky GN, Deliagina T, Grillner S. Neuronal Control of Locomotion: From Mollusc to Man. New York, NY, USA: Anonymous Oxford University Press; 1999.
42. Takahashi H, Takada Y, Nagai N, Urano T, Takada A. Serotonergic neurons projecting to hippocampus activate locomotion. Brain Research. 2000;869(1-2):194–202.[PubMed]
43. Iacono RP, Kuniyoshi SM, Ahlman JR, Zimmerman GJ, Maeda G, Pearlstein RD. Concentrations of indoleamine metabolic intermediates in the ventricular cerebrospinal fluid of advanced Parkinson’s patients with severe postural instability and gait disorders. Journal of Neural Transmission. 1997;104(4-5):451–459.[PubMed]
44. Cappellini G, Ivanenko YP, Poppele RE, Lacquaniti F. Motor patterns in human walking and running. Journal of Neurophysiology. 2006;95(6):3426–3437.[PubMed]
45. Dominici N, Ivanenko YP, Cappellini G, et al. Locomotor primitives in newborn babies and their development. Science. 2011;334(6058):997–999.[PubMed]
46. Smith SS. Step cycle-related oscillatory properties of inferior olivary neurons recorded in ensembles. Neuroscience. 1997;82(1):69–81.[PubMed]
47. Marlinski V, Nilaweera WU, Zelenin PV, Sirota MG, Beloozerova IN. Signals from the ventrolateral thalamus to the motor cortex during locomotion. Journal of Neurophysiology. 2012;107(1):455–472.[PMC free article][PubMed]
48. Collins SH, Adamczyk PG, Kuo AD. Dynamic arm swinging in human walking. Proceedings of the Royal Society B. 2009;276(1673):3679–3688.[PMC free article][PubMed]
49. Iannarilli F, Vannozzi G, Iosa M, Pesce C, Caprinica L. Effects of task complexity on rhythmic reproduction performance in adults. Human Movement Science. 2013;32(1):203–213.[PubMed]
The Golden Ratio and Our World
Leonardo of Pisa, better known as Fibonacci, was born in Pisa, Italy, about 1175 AD. He was known as the greatest mathematician of the middle ages. Completed in 1202, Fibonacci wrote a book titled Liber abaci on how to do arithmetic in the decimal system. Although it was Fibonacci himself that discovered the sequence of numbers, it was French mathematician, Edouard Lucas who gave the actual name of "Fibonacci numbers" to the series of numbers that was first mentioned by Fibonacci in his book. Since this discovery, it has been shown that Fibonacci numbers can be seen in a variety of things today.
He began the sequence with 0,1,… and then calculated each successive number from the sum of the previous two. This sequence of numbers is called the Fibonacci Sequence. The Fibonacci numbers are interesting in that they occur throughout both nature and art. Especially of interest is what occurs when we look at the ratios of successive numbers. The Fibonacci numbers play a significant role in nature and in art and architecture. When you construct a set of rectangles using the sequence (1, 1, 2, 3, 5, 8, 13, 21,), a design found in nature is revealed:
Next, when you construct in each square an arc of a circle with a radius the size of the edge of each respective square (a quarter circle), the organic design, which can be found in a snail shell can be seen:
Throughout history the length to width ratio for rectangles was one to 1.61803 39887 49894 84820. This ratio has always been considered most pleasing to the eye. This ratio was named the golden ratio by the Greeks. In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias. The space between the collumns form golden rectangles. There are golden rectangles throughout this structure which is found in Athens, Greece.
He sculpted many things including the bands of sculpture that run above the columns of the Parthenon. Phidias widely used the golden ratio in his works of sculpture. The exterior dimensions of the Parthenon in Athens, built in about 440BC, form a perfect golden rectangle.
Many artists who lived after Phidias have used this proportion. Piet Mondrian and Leonardo da Vinci both thought that art should manifest itself in continuous
movement and beauty. Therefore, they both expressed movement by incorporating the golden rectangle into their paintings. The golden ratio expresses movement because it keeps on spiraling to infinity. They showed beauty in their paintings by using the golden ratio because it is pleasing to the eye. To express the Fibonacci Sequence in art one must pay close attention to beauty, proportions, and continuous rhythm.
Leonardo Da Vinci dubbed this proportion the “divine proportion.” If you draw a rectangle around Mona Lisa’s face, you would find that the rectangle is in the golden proportion. He did an entire exploration of the human body and the ratios of the lengths of various body parts.
A modern day artist that used the golden ratio in a numerous amount of paintings was Mondrian. Piet Mondrian avoided any suggestion of reproducing the material world. Instead using horizontal and vertical black lines that outline blocks of pure white, red, blue or yellow, he expressed his conception of ultimate harmony and equilibrium. His style, and its underlying artistic principles, he called neoplasticism. Here is an example of one of his angular paintings which employ the proportion:
Composition with Gray and Light Brown
by Piet Mondrian
1918 (170 Kb);
Oil on canvas, 80.2 x 49.9 cm;
Museum of Fine Arts, Houston, Texas
Word Count: 606