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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? Written history vs. mammalian evolution2,000 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.

5,547 cm<sup>3</sup>... 5,547 cm is about the height of the Leaning Tower of Pisa.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!

JM Ruby
Roots and Routes

Originally posted to Ruby JM on Wed Jan 08, 2014 at 10:15 PM PST.

Also republished by SciTech and Community Spotlight.

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Comment Preferences

  •  Interesting. (7+ / 0-)

    How do kids react to this? It sounds pretty abstract. The time line seems more intuitive, at least for the cases where it fits in the room.
    My daughters school uses the timeline in a lot of different contexts. The students draw and illustrate the timeline of the dinosaurs, and of the roman empire, and of each individual students life. The kids see that it's a useful way to organize information.
    The time cube might be a good way for older kids to visualize some really big abstract concepts, but it's not as hands on as something the student draws themselves.

    Here's your horoscope for today: The universe doesn't even know that you exist.--Jbou

    by greycat on Wed Jan 08, 2014 at 10:45:04 PM PST

    •  Different tools for different purposes (13+ / 0-)

      1D timelines are definitely better suited for organizing information like you describe, especially with many data points (Julius Caesar killed at -43, Constantine's conversion at +312, with a dozen or so points between... same for millions of years, or the days of a pivotal event). When it fits in the room, I use it, even to the point of taking advantage of the hallway space!

      What my students do with time solids is visualize and write about differences in scale, like above, that do have exponential differences. They have made cubes the size of their lifetimes and put them within a larger cube to show the large difference between that and the millennia long timescales we study in an ancient history course that starts with the Neolithic Revolution. Once the cubes (or liter bottles, or whatever) are made, and placed within a room, manipulating them by stacking or pouring water or sugar gives them a sense of scale that they've reflected is missing when they just talk about years, millennia, and million-year periods.

      This is in a early high school context, but one where students are coming with low historical literacy, very little knowledge of exponentiation and logarithms, and interests that get piqued much quicker through physical manipulation than through figuring out there are a lot of zeroes involved. There's a sense of wonder in finding out how small your 8 sugar cube sized life is compared to the classroom that can represent the dinosaurs' and mammals' era combined.

      •  I always thought of time as 4D (1+ / 0-)
        Recommended by:

        One dimension of vector with a fixed point implying stasis or being at rest, one of space, one of volume, one of time. That way you can have a linear vector as the side of a space or membrane and the push pull sketchup as the side of a volume, and if you feel like it you can spin out a sphere or a cone or whatever shape you like ballistic so that it moves over time and so do all its nurbs and splines, with various rotations, movements, copies, arrays as necessary.

        What's interesting about that is that in 4d you can solve a lot of 3d problems without running into irrationals. People began doing this in the bronze age to arrive at concepts like gravity being related to matter and energy, space and time.

        Live Free or Die --- Investigate, Incarcerate

        by rktect on Thu Jan 09, 2014 at 07:07:09 PM PST

        [ Parent ]

  •  Another way to appreciate time (9+ / 0-)

    Wherever our minds may take us, we are always in the now.

    If cats could blog, they wouldn't

    by crystal eyes on Wed Jan 08, 2014 at 11:46:37 PM PST

  •  Just wonderful! (5+ / 0-)

    Not just the idea but the way you presented it. The video is dynamite.

    Yours are some lucky students.  

  •  Fascinating, and well-presented. (5+ / 0-)

    Hope the Rescue Rangers feature this.

  •  Very interesting! (4+ / 0-)

    Back in the 1980's, some of our displays at the Science Museum, London used similar symbology.

    Sadly, I was brought up in the era of log scales on the x and y axes of graphs!



  •  nice, so much more desired (3+ / 0-)
    Recommended by:
    Ruby JM, JVolvo, RiveroftheWest

    Not desired of you, necessarily. There are so many things about relative scale that I still haven't absorbed and wish I felt more deeply and less logically. Like the Eames' Powers of Ten stuff.

    Good job so far.

  •  Glad to see... (5+ / 0-)

    Glad to see both our science diaries in the community spotlight!

    FREEDOM ISN'T FREE: That's why we pay taxes! Find me on Linkedin.

    by mole333 on Thu Jan 09, 2014 at 09:01:21 AM PST

  •  Nice (6+ / 0-)

    What startles me, really, is not that time is so vast, but that in context, time is very short.  

    I'm 30.  My lifetime is already an appreciable chunk of the history of the United States; about one part in 8.  A slice of pizza from a pie.

    But America is a young country.  What about all of recorded history?  Something like 5,000 years.  But I'm still about half a percent of it.  Not very much.  But you give everyone in a normal high school graduating class one 30-year chunk, and you've got the whole thing.  Very human-scale.

    Humanity itself shoots back 200,000 years, and 30 is looking much more paltry.  Something like one part in 10,000.  But I've been to concerts and hockey games with many more people in attendance, all in one room.  That room full of people has more years' worth of memories than could stretch out over the entire human timeline, back to the savannahs.

    Time for a big jump.  The Universe, you know, is 13.8 billion years old.  A lot more than 30.  Somewhere on the order of 500 million times more (if you let me play faster and looser with the math as we go back).  But even at this scale, it's not like 500 million to one is an unimaginable ratio.  I know how much a dollar is worth, and 500 million of them is a record Powerball jackpot.  A single person at the federal poverty line stands in that proportion (roughly) to total government spending in the United States.

    If you think about the 7 billion of us who are alive today, if we averaged 30 years each, we have a total of 210 billion years of experience, over 15 times the age of the Universe!  Current human memory is enough to have watched photons from the big bang travel unimpeded to the present day.  

    It doesn't work for any other scale I can think of.  The mass of the universe is on the order of 10^53 kg.  I'm between 10^1 kg and 10^2 kg.  One part in 10^51 to capture that scale; nothing in human experience matches up.  All of humanity, even generously giving us 100 kg each, gets you to 700 billion kg.  The Three Gorges Dam, one, human-made structure, gets you on the right scale.  A reasonable mountain outmasses the lot of us.  We're less than the mass of the world's navies.  Our own constructs!

    How about length?  Start at 2m tall, and you're trying to get to 10^26 meters, end to end.  Better than mass, but not human scale, yet.  You get to a 14 billion meter human tower, all standing on each other's shoulders.  It's only twice the length of America's roadways.  It doesn't begin to get to galactic scales.

    The universe is young.  We are small, old things.

    •  I love your examples (2+ / 0-)
      Recommended by:
      RiveroftheWest, petral

      I, too, am fascinated by the possibility. It's one of the reasons I like the 3D visualizations so much. They show that these time scales are comparable.

      It reminds me of the concept of extended (or multi-generational) memory. I'm your age, but since my grandfather, born in 1921, told me stories of his own grandparents' lifetimes stretching back before the Civil War, you could say that my "two degrees of separation" goes back to 1836. People alive today can connect in the same way all the way back to the American Revolution (and events of the same era that span the globe), which does a lot to explain how much cultural continuity there can be, how dense our webs of connection.

      I'm thinking of another video exploring numbers of biological ancestors for this reason. Your number of ancestors n generations ago, at 2n exceeds the actual population of the Earth within 30 generations. Close relative pairings, but also a lot of migration, explain how the distributions work... but in any case, we are quite young.

      May we live to an old age (and healthy)!

  •  Well, there's the web-famous Time Cube (3+ / 0-)
    Recommended by:
    Ruby JM, RiveroftheWest, linkage

    so maybe you should shy away from that name. if you're not familiar.

    Amongst the longest-reigning kings of crank websites.

    "I wish you luck on not hating your parents for mixing up such an unthinkable person." --The frickin´ HP--

    by McWaffle on Thu Jan 09, 2014 at 12:00:05 PM PST

    •  Thanks for the warning! (1+ / 0-)
      Recommended by:

      I'd been warned already, but I was kind of hoping people searching for that sort of time cube on YouTube would get distracted by my video. Not sure if that is good strategy, as I've heard the guy is wrathful and he might think I'm an agent of some kind.

  •  Excellent! I think you're onto something (4+ / 0-)

    here.  Much more intuitive for large differences in relative magnitudes.  I'm awarding you an impromptu Edward Tufte pin!

    Almost nothing has a name.

    by johanus on Thu Jan 09, 2014 at 12:44:04 PM PST

  •  I suspect that a lot of DailyKos readers (1+ / 0-)
    Recommended by:
    Alice in Florida

    are going to be less than impressed until you get the representations up to at least 11-D format.

  •  Please continue to post your findings with regard (2+ / 0-)
    Recommended by:
    Ruby JM, RiveroftheWest

    to economics. politics, etc.

    I read an interesting article about the political and economical reach of Antioch before the Roman Empire absorbed it.  The research was based on the discovery of coins from Antioch around their region. Chronology was a facet under consideration.

    The link is here.

    •  That's really cool, I wish I could see the KMZs (1+ / 0-)
      Recommended by:

      Thank you. That's very cool how the decline of Seleucid power is actually visible in the YouTube clip based on her work. You don't get that detailed narrative without the animated map, because a narrative history isn't going to tell you about local decisions on currency like these maps too, and it won't be as detailed. Reminds me of how many Abbasid coins showed up in places like islands in the Baltic Sea.

      This a great reminder that data-centered archaeology isn't   just for prehistory. It can actually overturn conventional understandings of eras based on classicists and other historians who have tended to focus on literary narratives and epigraphy. Sometimes data illuminates more than the succession of rulers in the eventual winning dynasties and city-states that wind up in our textbooks!

      Have you seen the one about how coffee's reach may have fueled the Enlightenment?

      And thank you for the rec and encouragement!

  •  Enjoyed this. An interesting idea, looks useful. (2+ / 0-)
    Recommended by:
    Ruby JM, RiveroftheWest

    When a kid I counted to 100. Then repeated that count 10 times. So I could 'get' what '1000' referred to.

    From there imagining doing that '100 ten times' 100 times itself, and so on... that helped me grasp big numbers.

    But it still fails to feel meaningful with cosmic and microcosmic numbers. I could see how the 3D demonstration would generate a real appreciation for scale.

    Though I'm not sure where I'd put shipping containers in my home, or a school room.

    Actual Democrats: the surest, quickest, route to More Democrats. And actually addressing our various emergencies.

    by Jim P on Thu Jan 09, 2014 at 07:37:05 PM PST

  •  Thanks! (2+ / 0-)
    Recommended by:
    Ruby JM, RiveroftheWest

    This is so awesome. I will be incorporating this approach into lessons.

    Success is the sum of small efforts, repeated day in and day out. --Robert Collier

    by revm3up on Thu Jan 09, 2014 at 11:06:18 PM PST

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