You’ve been lied to since kindergarten.
Okay, maybe "lied to" is a bit dramatic, but your first-grade teacher definitely oversimplified things when they handed you that wooden block and said, "This is a square." We grow up thinking a square is just a square. Four equal sides. Four right angles. Done. But when you actually dig into the geometry of the world—and how we use these forms in design, architecture, and even digital interfaces—the idea of different shapes of squares starts to get a lot more interesting. It turns out that what we call a square is often just a starting point for a whole family of quadrilaterals that trick our eyes and solve complex engineering problems.
The "Perfect" Square Is a Mathematical Myth
In a purely Euclidean sense, a square is rigid. It’s defined by $a = b = c = d$ and $\theta = 90^\circ$. In the real world? Perfection is rare. If you look at a "square" tile on your bathroom floor through a magnifying glass, or measure it with a laser, it’s probably off by a hair. This brings us to the first major variation: the imperfect square.
Architects deal with this constantly. When you're building a house, foundation settling and material expansion mean that a "square" room is almost never actually square. Builders use the 3-4-5 rule (a tip of the hat to Pythagoras) to try and get close, but the reality is that most of the squares we live inside are actually very slightly skewed rectangles or rhombuses. It’s a game of perception. Our brains are hardwired to "correct" these shapes. We want to see a square, so we do. This psychological phenomenon is called shape constancy. It’s why you can look at a square table from an angle—where it technically looks like a trapezoid to your retina—and your brain still registers it as a square.
The Rise of the Squircle
If you’ve looked at an iPhone icon lately, you aren’t looking at a square. You’re also not looking at a rounded rectangle. You’re looking at a squircle.
This is probably the most famous of the "different shapes of squares" in the modern tech era. A standard rounded rectangle (or "roundrect") has a straight side that abruptly hits a circular arc at the corner. The human eye is actually pretty good at spotting that tangent point where the straight line turns into a curve. It looks a bit "clunky" to designers. A squircle, however, uses a mathematical formula—specifically a Lamé curve—to create a continuous, gradual transition from the flat side to the corner.
$$|x/a|^n + |y/b|^n = 1$$
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When $n=4$ and $a=b$, you get that beautiful, organic shape that feels "squarish" but much softer. Apple’s design team, famously led by Jony Ive for years, obsessed over this. It’s why their hardware and software feel so cohesive. It’s a square that breathed in and held its breath.
When a Square Becomes a Rhombus (and Why It Matters)
Here’s a fun fact that trips people up: every square is a rhombus, but not every rhombus is a square.
Think of a rhombus as a square that’s having a bit of a mid-life crisis. It’s lost its "upright" nature. All four sides remain equal in length, but the angles have shifted. This is more than just a geometry quiz answer; it’s a fundamental principle in structural engineering. When you see a "square" grid on a bridge or a fence, it’s vulnerable to shear forces. If you push on the top, it collapses into a flatter rhombus.
This is why engineers add a diagonal cross-brace. By turning that square into two triangles, they freeze the "square shape" in place. If you ever see a square-shaped gate sagging in someone's backyard, it’s because the wood has naturally shifted into a non-right-angled rhombus.
The Diamond Paradox
Is a diamond a different shape? Not really. In most contexts, a "diamond" is just a square rotated 45 degrees. But in the world of road signs, that rotation changes everything. A square sign (like a "Speed Limit") denotes a regulation. Rotate it 45 degrees into a "diamond," and it becomes a warning sign (like "Deer Crossing"). Same four equal sides, but the orientation changes the human psychological response. We perceive the diamond as more "dynamic" or "alert-inducing" because it balances on a single point, which feels inherently less stable than a square sitting on a flat base.
Tiling and the Geometry of Flooring
If you want to see how squares interact with other shapes, look down. Tessellation is the art of covering a surface with geometric shapes so that there are no gaps or overlaps. Squares are one of the only regular polygons that can do this perfectly on their own (the others being triangles and hexagons).
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But designers often use truncated squares. Imagine taking a square and snipping off the corners. If you snip them just right, you get an octagon. In many classic Victorian floor patterns, you’ll see large octagons (truncated squares) interspersed with tiny "square" tiles (called tozzettos). It’s a clever way to use the DNA of a square to create something that looks much more complex.
Then you have quadrille patterns. This is basically a grid of squares, but it’s often used in fabrics and wallpapers to create a sense of order. When you vary the line weights of these squares, you get graph paper; when you fill them with alternating colors, you get a chessboard.
The Square in Three Dimensions: The Cube and Beyond
We can’t really talk about different shapes of squares without mentioning what happens when they grow a third dimension. A cube is the obvious evolution. But in advanced geometry, we talk about tesseracts (four-dimensional hypercubes).
While we can't fully visualize a 4D shape, we can see its "shadow." If you unfold a 3D cube, you get a 2D cross shape made of six squares. If you "unfold" a 4D tesseract, you get a 3D cross made of eight cubes. It’s a bit of a brain-melter, but it shows that the square is the foundation for understanding higher dimensions of reality.
Common Misconceptions About Squarish Shapes
Most people use the word "square" as a catch-all for any rectangle that looks "close enough." But accuracy matters in specific fields:
- The Aspect Ratio Fallacy: In digital photography, we talk about "square format" (1:1). Many people think Instagram photos are still strictly square, but the platform moved away from that years ago. Most "squares" you see on social media are actually 4:5 ratios.
- The "Boxy" Car: When we call a car like the old Scion xB or a Mercedes G-Wagon "square," they aren't. They are series of complex rectangles and trapezoids. True squares are actually quite aerodynamically inefficient, which is why you rarely see them in motion-based engineering.
- The Quadrilateral Hierarchy: People often forget that a square is the "gold standard" of quadrilaterals. It's the only one that is both equilateral (all sides equal) and equiangular (all angles equal).
How to Use "Square" Logic in Your Life
Understanding the different variations of this shape isn't just for math nerds. You can actually apply this to how you organize your world.
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In Home Design
If you have a small room, using square rugs or square coffee tables can actually make the space feel more static and "stuck." To create movement, you want to break that squareness. Use a "squircle" table—one with rounded edges. It allows for better traffic flow because you aren't constantly dodging sharp 90-degree corners.
In Photography
The square crop is powerful because it removes the "horizon" bias. A horizontal rectangle feels like a landscape; a vertical one feels like a portrait. A square? It centers the subject perfectly. If you’re trying to highlight a specific object (like a plate of food or a single flower), crop it into a perfect square. It forces the viewer’s eye to stay put.
In Web Design
Stop using sharp-edged squares for buttons. Research in user interface (UI) design consistently shows that humans find rounded corners (those squircles again!) easier on the eyes. Sharp corners point away from the center of the object, while curved corners lead the eye toward the content inside the button.
Real-World Expert Insight: The Bauhaus Influence
The famous Bauhaus school of design in Germany obsessed over the square. To them, the square represented the modern, industrial man. Wassily Kandinsky, a teacher at the school, actually associated the square with the color red—feeling it was a heavy, grounded shape. When you look at modern minimalist architecture, you're seeing the legacy of people who thought deeply about whether a window should be a perfect square or a "golden rectangle."
Next Steps for Mastering Shapes
You don't need a degree in geometry to appreciate the nuances of the shapes around you. Start by looking at the "squares" in your immediate environment. Take a ruler to that "square" picture frame on your wall—is it actually 10x10, or is it 10x10.2?
If you're a DIYer or a crafter, practice "squaring up" your projects using the diagonal method. Measure from the top-left corner to the bottom-right, then from the top-right to the bottom-left. If those two numbers are identical, you’ve achieved the "perfect square." If they aren't, you've got a rhombus.
For those interested in the digital side, try downloading a vector tool like Figma or Adobe Illustrator. Experiment with the "corner radius" tool to see at what point a square stops feeling like a box and starts feeling like a button. Understanding these subtle shifts in geometry is the first step toward having a "designer’s eye."