Fourth Dimension

Today, the missing dimensions.  The University of Houston presents this series about the machines that make our civilization run, and the people whose ingenuity created them. 

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     It’s 1951: I’m a twenty-one-year-old neophyte engineer.  And an old man befriends me – a retired drafting instructor.  Now: drafting’s a subject I really loved. Taking three-dimensional objects out of my head and displaying them on two-dimensional paper.

     But this fellow has contrived means for drafting four dimensional objects on two-dimensional paper!  So I’m fascinated.  Take the progression from a 2-D square to a 3-D cube to a 4-D what? Well, we have a name for that.  A 4-D cube is called a tesseract. The 3-D view from one side would be a cube.  But look at it from an angle, and you get a very weird 3-D object. 

     This man has written a beautifully illustrated book showing how to draft 4-D objects.  But ... and who is surprised ... no one wants to publish it.  So forget drafting and think about our lives.  Imagine our entire life, from birth ‘til now, displayed in the four dimensions of space and time.  That four dimensional record exists; but all we can ever see of it is this 3-D moment.

     Go online and we find a dizzying lot of pictures of tesseracts. But none are very helpful since our minds can’t really make the jump from a wiggly image on a two-dimensional screen to four dimensionality.  And yet: Our science and engineering are laced with multidimensional realities. 

Try this: Imagine that we want to describe what happens to water when we change its pressure, temperature, and density.  Fine: We plot it in those three dimensions.  And we get a nice visual picture of it.  Specify a pressure and temperature, and we immediately see what the density is.

But -- suppose we want to do the same thing with various mixtures of alcohol and water.  Now our graph would have to be four dimensional.  And, unless I were to appeal to my old friend’s drafting book, I have to rely on separate graphs for different portions of water and alcohol.

A 3-D representation of Pressure, Temperature and composition in an alcohol-water mixture.  

(But what is the density at any point?)

     Or (and here’s where this problem takes us) — we can write an equation that includes all four properties – density, temperature, pressure, and composition.  In the end, mathematics can tell us what our eyes cannot. 

     Equations that employ four or more variables are commonplace. My own field of research is heat transfer.  And there the rate of cooling an object usually depends upon far more than two variables – temperatures, physical properties, size ... 

So, in the end, the world that my old friend wanted to display, really is accessible.  A vast world opens up to us, but not on his drawing board.  It opens through the amazing utility of mathematics. Mathematics, it turns out, can tell us far more than we could ever see in our mind’s eye.

I’m John Lienhard, at the University of Houston, where we’re interested in the way inventive minds work. 

(Theme music)

Here is the Wikipedia page about four-dimensional space.

Here Wikipedia explains the old skill of drafting.

My “A Heat Transfer Textbook” readable online is shot through with equations that depend upon far more than just two variables.

This Episode first aired on June 29, 2026.