Have you ever wondered how we know we live in a three-dimensional world? It’s not just because we use three coordinates like x, y, and z (or latitude, longitude, and altitude) to describe locations in space. Surprisingly, mathematicians have discovered that it’s possible to fill up two-dimensional or three-dimensional space using a one-dimensional “space-filling” curve. This means that every point in 3D space can be labeled with just one coordinate, which is our position along this curve. It’s a bit mind-blowing, like realizing a square and its side have the same number of points!
So, how do we know we live in three-dimensional space and not on a one-dimensional line that’s so twisted it appears three-dimensional? The truth is, we can’t be absolutely sure. However, we do know that our world looks 3D to us.
One way to test the dimensionality of our world is by observing how gas diffuses or spreads out over time. By measuring the ratio between the volume and the radius of a gas cloud, we can gather clues about the dimensions we live in. In one dimension, the radius and volume are essentially the same, differing only by a factor. In two dimensions, “volume” refers to the area, which is the radius squared. In three dimensions, “volume” is the radius cubed. This pattern continues for higher dimensions, but what we observe in our world is consistent with three dimensions.
In essence, determining the number of dimensions we live in might seem like a lot of “hot air,” but it’s a fascinating exploration of how we perceive and understand the space around us. While we can’t be entirely certain, the evidence strongly suggests that our world is three-dimensional, and that’s how we experience it every day.
Use modeling clay or 3D modeling software to create a representation of a three-dimensional object. Consider how you would describe this object using three coordinates. Reflect on how this model helps you understand the concept of living in a three-dimensional world.
Research and draw a space-filling curve, such as the Hilbert curve. Try to visualize how this one-dimensional curve can fill a two-dimensional or three-dimensional space. Discuss with your classmates how this concept challenges our understanding of dimensions.
Perform a simple experiment to observe gas diffusion. Use a balloon filled with a scented gas and measure how the scent spreads over time. Analyze the results to understand how diffusion provides evidence of three-dimensional space.
Create an art piece that represents different dimensions. Use various materials to depict one-dimensional, two-dimensional, and three-dimensional spaces. Present your artwork to the class and explain how each piece represents its respective dimension.
Participate in a class debate about whether we can be certain that we live in a three-dimensional world. Use evidence from the article and other sources to support your argument. Consider the implications of living in a world with different dimensions.
Three-dimensional – Having or appearing to have length, breadth, and depth – In physics, a three-dimensional space is often represented by the x, y, and z axes.
Coordinates – A set of values that show an exact position in a space – The coordinates of the point where the two lines intersect are (3, 4, 5) in three-dimensional space.
Space – A boundless, three-dimensional extent in which objects and events occur and have relative position and direction – Mathematicians often study the properties of different shapes within Euclidean space.
Mathematicians – Experts in or students of mathematics – Mathematicians use complex equations to model the behavior of physical systems.
Curve – A smoothly flowing, continuous line or surface that differs from a straight line in any way – The curve of the graph represents the trajectory of the projectile in motion.
Volume – The amount of space that a substance or object occupies – Calculating the volume of a cylinder involves using the formula V = πr²h.
Radius – A straight line from the center to the circumference of a circle or sphere – The radius of the sphere is crucial for determining its volume and surface area.
Area – The extent or measurement of a surface – To find the area of a rectangle, multiply its length by its width.
Dimensions – Measurements in width, depth, and height – The dimensions of the box are needed to calculate its volume.
Gas – A state of matter consisting of particles that have neither a defined volume nor shape – The ideal gas law relates the pressure, volume, and temperature of a gas.
