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Fractal Matrix @ Mandelbrot Fractal

The web of our life is of a mingled yarn,...

...good and ill together.

William Shakespeare
All's Well That Ends Well [1601-1603]

Selected Portfolio

Past Forward


Research Consultant

, circa 2005
...National Science Foundation Research Nugget

Achieving direct tactile feedback similar to the biological sense of touch is a major challenge in robotic sophistication. Opening new areas of exploration in remote sensing systems, a team of UIUC engineers — graduate students Jonathan Engel and Jack Chen, senior research engineer Zhifang Fan, and associate professor Chang Liu — has developed a flexible and robust polymer-based tactile "skin" inspired by the biological model. Using simple, thin-metal film building blocks and a variety of physical probes, this first generation design "senses" the hardness, temperature, and thermal conductivity of materials. The tactile sensor is capable of direct contact with everyday objects and could serve as a discriminating "skin" on robotic actuators.


Enterprise Webmaster

, circa 2000
...University Newsletter Request

Please enter the following information to submit your news-worthy item as you would like it to appear. All submitted copy is subject to editing. You will be contacted by our office after your submission has been received. Items will be accepted until 4:30 p.m. on Monday of each week for the newsletter's e-mail notification and web distribution on Tuesday.


Sending your message to our office requires a two-step process. When you press the Continue... button below you will have the opportunity to proof the content of your message. You will then have the opportunity to press a Confirm button to send the information to our office or to press the Back button to modify your message. Until you press the Confirm button, your message will not have been sent to our office. After pressing the Confirm button on the next page you will also have the opportunity to print a copy of the information that we receive. You will be contacted by our office when the submission has been received.


College Educator

, circa 1995
...Rationale for Cooperative Group Learning

Learning is a three step process—presentation, practice, and performance. The presentation is where someone shows you how to do something. The practice is where you show yourself how to do something. And most important, the performance is where you must show someone else how to do something. The need for this last and often neglected phase is based on studies of long-term learning:

  • students learn...95%...of what they show to someone else,
  • students learn...80%...of what they experience,
  • students learn...70%...of what they discuss,
  • students learn...50%...of what they see and hear,
  • students learn...30%...of what they see,
  • students learn...20%...of what they hear,
  • students learn...10%...of what they read.

Balanced Learning The three phases of learning may be visualized as a triangle where the length of each side indicates the emphasis on any one phase. Balanced learning would correspond to a triangle with three equal sides. To achieve the highest success—to show, to experience, and to discuss—we will advocate...


Research Scientist

, circa 1985
...Article Published in the J. Amer. Chem. Soc.

Structures of fractional dimension are known. Mandelbrot4,5 has pioneered the theoretical concepts and physical applications of the relatively new field of geometry6,7 and has popularized the term fractal for a structure characterized by a fractional dimension. By definition, any structure possessing a self-similar or repeating motif that is invariant under a transformation of scale may be represented by a fractal dimension. Self-similarity is geometric in regular structures; in random or irregular objects, self-similarity is primarily statistical in nature. The average (root mean square, RMS) end-to-end length R of an unbranched polymer chain constitutes a statistically self-similar property. The fundamental relationship in fractals predicts that the number of monomer segments N of length r is related to R by the fractal dimension d

N = (R/r)1/v ≡ (R/r)d    (3)

where the exponent is also equal to the inverse Flory constant v in polymer theory8. Theoretical considerations provide limits of 1 ≤ d < 2 in eq 3 that correspond to a linear structure, R = rN, and a structure represented by an unrestricted random walk, i.e., Brownian motion in d ≥ 2, where R = rN1/2.9

Past Forward...

The web of our life is of a mingled yarn,...

...good and ill together.

William Shakespeare
All's Well That Ends Well [1601-1603]





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