Fractals – Geometry Within Us

Biological Sciences


"I find the ideas in the fractals, both as a body of knowledge and as a metaphor, an incredibly important way of looking at the world."
Vice President and Nobel Laureate Al Gore

This WebQuest is intended to have learners use collaboration, creativity, and internet resources to see how human body structures and especially lungs can be explained through fractal geometry.

Do you know that there is something in common between your own body, galaxy structures, and cauliflower? Have you ever wondered how rivers resemble trees that resemble our blood vessels? In this WebQuest, we are going to take a journey during which we will explore these questions discovering some beautiful structures called fractals.

But before that, here are a few things you will need to know:


The goal of this WebQuest is to help you discover the importance of making interconnections between different areas of knowledge. In the case of this WebQuest, they will be mainly biology (with special attention to anatomy and lungs) and mathematics (fractal geometry).

So, let’s start our investigation and have a look at the beautiful fractal universe within us.

The learners should work in groups of 5 to 10 people. They should explore the concept of fractals and discover their structure within the human lungs. Each of them should prepare a short PP presentation (or other multimedia) on another part of the human body having a fractal structure (blood vessels, kidneys, etc.) The information should be well organized, factual, and well-formed. It is important to demonstrate a good level of internet and technology use


At this step, each learner should use the online resources for this Webquest and research the nature of fractals and their main characteristics.

They can start with the following definitions:

Basic information on fractals has been well summarized here:

In the following videos the man who discovered fractals and invented the word fractal itself, Benoit Mandelbrot, explains their nature in a concise and inspiring way:

So, true fractals often have clear systems at endlessly small scales. Basic fractals are self-similar and due to this, they are said to be immeasurably complex. Examples of some real fractals are coastlines, cauliflower, snowflakes, clouds, mountains … and many organs of our bodies. Fractals can be also computer-generated. However, computers repeat impeccable copies in an infinite way. That never happens in nature and indeed nature takes advantage of that fact. For example, if our hearts were beating with the same intensity all the time, they would not have a chance to relatively rest from time to time.

Indeed, full symmetry or perfect repetition regarding the human body is rather a sign of some kind of disease…

By the way, have you noticed how much a cardiogram resembles a coastline? Well, that is due to their fractal structure.

So, great, folks! You already know what fractals are and looking at the picture on the left it’s obvious to you that lungs are essentially fractal even if you don’t know much about their structure or the way they function. You can now imagine how you inhale oxygen and follow its way through the fractal tree-like structure of your lungs. …or just watch how it happens in this beautiful video:

These websites will help further realize the lungs’ fractal nature:

Our lungs are branching fractals with a surface area of ~100 m2. Their similarity to a tree is significant, as lungs and trees both use their large surface areas to exchange oxygen and CO2. If you let me cite Mr. Mandelbrot from the video above: Now a lung is something very strange. If you take this thing, you know very well it weighs very little. The volume of a lung is very small, but what about the area of the lung? Anatomists were arguing very much about that. Some say that a normal male’s lung has an area of the inside of a basketball [court]. And the others say, no, five basketball [courts]. Enormous disagreements. Why so? Because, in fact, the area of the lung is something very ill-defined. The bronchi branch, branch, branch and they stop branching, not because of any matter of principle, but because of physical considerations: the mucus, which is in the lung. So what happens is that in a way you have a much bigger lung, but it branches and branches down…

The lungs might look small when you look at its outer surface area but inside they are much larger. This is because the lungs grow as branching fractals. Watch this video and see how the lungs develop in the embryonal phase:

As you have seen, lungs develop like fractals and show the properties of fractal geometry. But due to their fractal nature, lungs are also more efficient. The alveoli are the little air sacs in the lungs whose main task is to circulate oxygen in the blood. The time at which the circulation happens and the rate at which the diffusion of air in the alveoli occurs is directly proportional to the area of the alveoli in the lungs.

Here you can have a close 3D look at human lungs and see how they function in some short videos:

The process of breathing is further explained here:

So, finally, have a look at this website that summarises some of the interesting facts that you already know about lungs:

At this stage, learners should explore other fractal structures of the human body. This area of study is still a developing one and they will have to make their own investigation. They can explore kidneys, brain, DNA, heart, neurons…These websites can provide further ideas:

Come on, it’s up to you to discover the beauty of fractals somewhere within your body!

Use multimedia (PowerPoint, Prezi, other tools) to present the results of your investigation on fractal structures in the human body. You can include pictures or videos. Don’t forget that one of the traits of Mr. Mandelbrot’s fractal geometry is its visibility.

At this stage, learners should discuss the presentations and comment on some other fractal structures in the human body and nature they can think of.


Through this WebQuest you experienced the fact that human body is fundamentally fractal.

Thus, it follows basic structures that dominate our world and the universe as a whole on different scales.

Most people find fractals relaxing and beautiful. Please, watch this video (or just a part of it) and discover if you find them soothing, too:

If so, discuss why. Could it be because we unconsciously discover a fundamental principle of the structure of our own body in the fractals we watch? Anyway, by analyzing the nature of beauty we are already in the realm of aesthetics, which is yet another area of knowledge.


  • Presentation skills
  • Critical thinking
  • Research


In addition, learners will:

  • realize that areas of knowledge are interconnected and this is crucial for their development;
  • learn how to engage and collaborate with others;
  • communicate an idea by means of a discussion;
  • develop a sense of confidence and belief in themselves and their ideas;
  • learn how to communicate effectively (using written and spoken word, non-verbal language, electronic tools, and listening skills).

Evaluation of learning achievements

In this section we will not dive very deep into the underlying educational theories about evaluation and testing: there’s too much out there than we could possibly cover in this small project report.

Instead, we want to concentrate on procedures that enable both students/pupils and their teachers to establish if the learning goals of the Webquest were achieved and, if so, to what extent. We recommend teachers make use of a combined evaluation procedure, that consists of:

  1. Statements by learners (after being asked to do so)
    • telling what they learned about the subject (knowledge-oriented self-evaluation): now (after going through the Webquest) I know that …
    • telling what he/she learned about herself/himself (formative evaluation, in this case, diagnostic self-evaluation): now (after going through the Webquest) I know about myself that I …
      This pair of basic statements add up to a so-called learner report, in which the pupil/student reflects on what the Webquest brought him/her in terms of acquired knowledge and new personal views and attitudes concerning the subject.

    For instance:

    • ‘I learned that in medieval times the hygiene of people was hardly a concern which helped to let epidemic diseases like the Plague cause so many casualties’ Or:
    • ‘I learned the facts and I know the earth is warming, but I cannot understand why people were so stupid to pollute the world and let it warm up so much.
    • ‘I learned from the information about diseases that this subject is more appealing to me than I would expect in advance: maybe I should consider a medical career’. Or:
      ‘The Webquests confirms what I thought already: I could not care less about the climate and global warming. In fact, I thought it was all a hoax and I still do!’

    This kind of assessment seems more subjective than it actually is: in his standard work on testing and evaluation (and much more), simply called Methodology (1974), Prof. A.D. de Groot described how consistent the student’s self-evaluations appeared to be: when asked again after 5 or 10 years, their evaluation would almost be the same. De Groot advised teachers to use the learner report as a start for joint evaluations, striving for consensus between teacher and student/pupil about the learning outcomes and their value for the learner, but also compared with the learning objectives as stated in the curriculum.

  2. The learning achievements are visible in the output produced by the students: it is physical evidence: reports, answers to questions asked in the Webquest, presentations, and performance during presentations (preferably recorded). The teacher completes an evaluation grid stating clearly what the learning outcomes for the student/pupil are. The categories in the grid can be modified by the teacher to cover more precisely the content of a Webquest.

    >We advise teachers to use the grid to start a joint evaluation discussion, aiming at consensus or at least understanding between the teacher and the student/pupil about the learning outcomes: were they achieved (as planned in the curriculum and communicated before the Webquest started) and to what extent? To communicate the learning goals clearly before any learning activity starts, is a transparency requirement that is widely acknowledged in the educational community. The history of making learning objectives explicit goes back to the evaluation ‘Bible’ by Bloom, Hastings and Madaus: ‘Handbook on formative and summative evaluation of student learning’ (1971), a standard work that also served as inspiration for the earlier mentioned Prof. De Groot.


The procedure also applies when students/pupils have worked together on a Webquest. The teacher will ask questions about individual contributions: ‘What did you find? What part did you write? How did you find the illustrations? Who made  the final presentation?’

All the evidence (of learning efforts and outcomes plus joint evaluations) is preferably stored in the learning portfolio of the student, or in any other suitable storage system (folders with written or printed documents, online collection of files, etcetera ).

Changes in personal points of view and feelings are harder to value and here the consensus between teacher and student/pupil about experiences during the learning process provides essential insights.

The grid below gives an example of how the evaluation of the learning process and achievements can be shaped: what kind of reactions to the Webquest does the teacher expect and how valuable are they? Is the teacher capable to explain the value or score allocated to answers or presentations given by pupils? Does the pupil/student understand the evaluation outcomes, and does he/she agree? If an agreement (consensus is not possible, it is still the teacher who decides how to value the student’s work.

Please note that the text in the grid addresses the pupil/student directly: this is important and it is in fact a prerequisite for using such an evaluation grid: it is specifically meant to enable a discussion of learning results between teacher and student and not to communicate learning achievements of learners to others who had no direct role in the Webquest.

Evaluation Grid

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Funded by
sCOOL-IT erasmus logo EN

The European Commission’s support for the production of this publication does not constitute an endorsement of the contents, which reflect the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

Talk To Us

t: +357 2466 40 40
f: +357 2465 00 90

Funded by
sCOOL-IT erasmus logo EN

The European Commission’s support for the production of this publication does not constitute an endorsement of the contents, which reflect the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

Talk To Us

t: +357 2466 40 40
f: +357 2465 00 90

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