# Incomprehensibly Small and Incomprehensibly Large

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Initial Publication Date: May 7, 2008 | Reviewed: November 2, 2013

#### Summary

Students are asked to consider the length scales of the universe, from smallest to largest. Using a logarithmic scale in cm units, various distances are resolved ranging from the smallest meaningful length in Nature (Planck length) to the largest (cosmic horizon). Students compute the lengths of several given distances, plot them on the log scale, and join in a discussion.

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## Context

#### Audience

I have used a version of this exercise during class in an introductory physical geology course that has no prerequisites.

#### Skills and concepts that students must have mastered

Students need to have a basic understanding of exponents (leading to scientific notation), which is reiterated at the beginning of the exercise.

#### How the activity is situated in the course

This activity is used near the beginning of the course as we are discussing the ranges of such things as length, time, velocity, temperature, and pressure that affect the physical reality we study in geology. The results can be referred to in subsequent discussions, so it is best to use this exercise early in the semester.

## Goals

#### Content/concepts goals for this activity

I want students to understand the relative sizes of a few key parts of physical reality.

#### Higher order thinking skills goals for this activity

This exercise involves some simple computing (e.g., 4.2 light years is equal to how many centimeters?), manipulation of exponents, plotting of data on a graph. Students are asked to work toward an understanding of a logarithmic scale, and what "orders of magnitude" means.

#### Other skills goals for this activity

This is usually done using small groups for computation and discussion, and the small groups share their results with the class. So skills related to group/team work are developed, it is hoped.

## Description of the activity/assignment

I usually begin with a story about lying on a cot looking up at the stars on a dark night in the mountains, seeing countless stars and the hazy Milky Way stretching across the sky. I talk about how they seem to be part of a celestial dome rising very high above me, and I note that I do not have any way to know, as I am looking at the stars above me, how far they are away from me. I talk about how ancient people used and envisioned the stars. I mention the experiment with the Hubble Space Telescope in which the "darkest" and most empty part of space was imaged, and found to contain countless distant galaxies (search on "Hubble deep field" or go to http://www.stsci.edu/ftp/science/hdf/hdf.html).

I mention that this often leads people to consider how insignificant they are in the scheme of things. My feeling is that you are only as significant (or insignificant) as your actions make you.

I then talk a bit about how we now know that "visible" matter is organized into atoms, which are very, very small. In a way, they are like the stars in that they seem to be incomprehensibly small, while stars seem to be incomprehensibly large and distant. I then pose the question, "How does the part of this world that we observe and experience on a daily basis fit into a physical reality that spans from the incomprehensibly small to the incomprehensibly large?"

I pass-out the blank worksheet "Comparison of Lengths Relevant to Our Universe" to every student, and have them organize into groups of 2-3. The task is to fill-in the exponents corresponding to 9 distances listed in a box on the page, and to locate those distances on the logarithmic scale. I give them a couple of minutes to start working with the page, and then interrupt to ask what they need help with. This usually involves determining one of the lengths involving light years on the board. I let them complete the tasks in their small groups, then I ask group representatives to call-out their results.

Working from a set of correct answers, we then discuss the scale. For example, we note that there is a greater difference (in orders of magnitude) between the size of a proton or electron versus the size of a hydrogen atom, and the height of a person and the peak elevation of Mt. Everest. It is usually noted that humans fall near the middle of the length spectrum of the universe, which was also noted by Primack and Abrams (2006). Some students place great importance on this. I tend to note that there is a practical limitation to the size of individual cells that will have predictable functions (they need to be larger than the length scale governed by quantum mechanics) and constraints on the upper size limit of organisms made of cells, which determines where we are on the scale.

## Determining whether students have met the goals

After students have worked on the handout, answers are shared with the class and the teacher eventually discloses the correct answers. Once the correct answers are held in common, further discussion about the information is encouraged.

## Teaching materials and tips

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• Solution Set (Acrobat (PDF) 448kB May5 08)

## Supporting references/URLs

Robert B. Laughlin, 2005, A different universe, reinventing physics from the bottom down: Cambridge, Massachusetts, Basic Books, 254 p., ISBN 0-465-03828-X

Joel R. Primack and Nancy E. Abrams, 2006, The view from the center of the universe: New York, Riverhead Books, 386 p., ISBN 1-59448-914-9

Wikipedia: Emergence

Hubble Deep Field

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