When asked to visually represent periods of time, we often turn to the venerable timeline.
Linear, one dimensional, and familiar, the timeline fills our history textbooks with precisely spaced chronologies, serves as the x-axis in graphs of real minimum wage or CO2 emissions over time, and helps us organize our iStuff. Yet for the purpose of comparing the short periods of time that we experience to evolutionary eras or to cosmological vastness, timelines utterly fail.
If we try a scale of 1 year per centimeter, our lifetimes fit comfortably within a meter, but the age of the Earth (4.54 billion centimeters (45,400 kilometers) exceeds the circumference of the Earth. If we try 2 million years per centimeter, the evolution of birds and mammals can fit on a single sheet of legal size paper, but human lifetimes shrink down to the nanometer scale and the age of the Universe requires 69 meters (226 feet).
But there is no reason to despair. As 3D beings we can touch, manipulate, and form 3D shapes. Therefore I propose, as an teaching aid, the use of time solids as a 3D alternative to 1D timelines, as the power of exponentiation makes them fit this purpose admirably.
3D objects grow n^3 times faster than 1D objects, and familiar size containers serve as great approximations for both long and short periods of time. A 1 centimeter scale works perfectly for this, showing the lifetime of a baby with a sugar cube, a 65 year old's lifetime with a cube with a 4 centimeter edge, 2,000 years with a 2 liter bottle, all of recorded history with a soccer ball, the time since the extinction of the dinosaurs with a 40' intermodal shipping container, and even the entire age of the Universe on a 110 meter by 67 meter soccer field, multiplied by a normal human height!
Please follow me below for the math and extended discussion. All of this is part of a channel (1st video, here) in which I'm attempting to show examples like these, and relate social studies, science, and math pedagogy in exciting ways. As a high school educator and enthusiastic nerd, I am very interested in ways that visualization and hands-on work can shape education.
1 cubic centimeter looks, unsurprisingly, similar in size to 1 centimeter.
As you scale up, one advantage for 3D visualizations of time quickly becomes apparent. My 31 year lifetime requires an entire ruler for 1D, but in 3D you only need a cube with a 3.14 cm (1.24 in) edge, or 31 grams (about a fluid ounce) of water. The time between now and the U.S. Declaration of Independence requires standard residential ceiling height for 1D, but in 3D cube with a 6.19 cm (2.44 in) edge suffices, or 238 grams (about 8 ounces) of water. If you don't want to make cubes or measure water in glasses, anything labeled in milliliters is a 1:1 conversion to years, since a milliliter is the same as a cubic centimeter. The 2 liter bottle in your fridge? 2,000 years, enough time to approximate the entire Common Era or, more precisely, to the final year of Augustus, first Roman Emperor. The gallon of milk? 3,785 years, to the reign of Hammurabi. Representing this time period with 1D centimeters requires 38 meters (125 feet). To represent the time since the age of the dinosaurs you'll need two 20' shipping containers (66.2 cubic meters equals 66.2 million cubic centimeters) or a 650 kilometer flight.
My video on these visualizations took me to a soccer field as filming site because a 69 centimeter circumference soccer ball has a volume of approximately 5,545 cubic centimeters, enough represent the time since 3532 BCE. The field itself, at a height of 62 centimeters (about 2 feet), is large enough to represent the 4.54 billion year age of the Earth and, at a height of 190 centimeters (about 6 foot 2), the 13.8 billion year age of the Universe! Billions of cubic centimeters are only thousands of cubic meters, and the Randall's Island Park Soccer Fields' 110 meters * 66 meters * 1.9 meters yields the 13,800 cubic meters (13.8 billion cubic centimeters) that we need. To represent the age of the Earth linearly, at 1 cm/year, you would need 45,400 kilometers—coincidentally, the approximate circumference of the Earth. For the age of the Universe, you'll need to go around the Earth three times, or take 25 New York-London flights. The image to the left shows the 6,000 year time sphere on its way to midfield, all of recorded history an insignificant volume compared to the field that represents age of the Universe. Yet, unlike the circumference of the Earth, we can experience the soccer ball, even carry it or kick it through every cubic meter (1,000,000 years) of space.
It even works the other way. A day, as 1/365th of a year, requires 2.74 cubic millimeters, or a cube with 1.39 mm edges.
It's tiny, compared to a soccer field, but visible. The soccer field is enormous, but human scale. We can walk, in not all that much time, within every one of those cubic centimeters, back and forth across the Universe field, kicking or carrying with us a time sphere the size of recorded history. 1D comparisons, of a ruler to the circumference of the planet, don't do us much good, because we don't experience both distances at human normative speeds. The speed at which we walk or drive is not the same as the field at which we cross oceans.
It works because of the power of exponentiation. 100 centimeters to 1 meter, but 100 * 100 * 100 = 1,000,000 cubic centimeters to 1 cubic meter.
There are, of course, some intuitive problems with 3D visualization. Objects closer to us have a much greater angular diameter and thus appear much larger than objects far away. The still image of the soccer ball on the field, seen in 2D, is thus still much too large. We don't intuitively understand how much faster volume increases than surface area or length, so perhaps this scale can make time periods seem insufficiently vast. Then again, who can mentally compare an inch to the Earth's circumference?
Time to time analogies are better. To reference Carl Sagan's classic Cosmic Calendar, if the age of the Universe is taken to be one year, with now being December 31 at midnight, the dinosaurs went extinct early yesterday, and all of recorded history is about10 seconds long. Logarithmic scales can capture vast and small times well, but since we don't perceive 2D surfaces logarithmically (unlike pitch or loudness), they are not quite intuitive. Video technology, like on ChronoZoom, can really help us see how small days and years are compared to the lifetimes of star systems.
All of these are beautiful ways of understanding. Yet the advantage of 3D is in its manipulability. Although it's hard for us to really understand that if there are two cubes, one with 5 times the edge length, that the larger cube is 125 times the volume, it becomes easier if we can count discrete objects. If I'm a middle school student who knows that a sugar cube represents about 2 years at this scale, my life is 6, my teacher's is 20, and the time since the building of the Giza Pyramids about 15 1 pound boxes worth, I can appreciate more about human history and learn a lot about exponents. I can even take the time to count out 2,250 sugar cubes (or quickly give up on that). Pouring masses of water at 1 gram per milliliter per year is even easier, a true mix of science, social studies, and math understanding. Classrooms can be measured too. Mine is a bit less than 300 million years in volume.
I am thinking that it could also work for understanding differences between thousands, millions, billions, and trillions for a little attempt at understanding the vast scale differences involved in economics. I'll be working on that, with my students, and will hopefully be able to report it here and on my YouTube channel.
Thank you very much for reading and viewing. This is such a valuable community of educators and thinking people, and I'm very much looking forward to discussing this, if I'm fortunate enough that it gets read!
Roots and Routes