Lecture 2: First Constructions and Platonic Solids
Formal Euclidean Geometry, Prof. Kontorovich
Rutgers Math Corps, Summer 2025
Recall: Euclid's Five Postulates
Before diving into our first construction, let's quickly review Euclid's fundamental assumptions:
Postulate 1
To draw a straight line from any point to any point.
Postulate 2
To produce a finite straight line continuously in a straight line.
Postulate 3
To describe a circle with any center and radius.
Postulate 4
That all right angles equal one another.
Postulate 5 (The Parallel Postulate)
That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Proposition I.1: Constructing an Equilateral Triangle
Euclid's First Proposition
Statement: To construct an equilateral triangle on a given finite straight line.
More formally: We are given points A and B. Our goal is to construct an equilateral triangle using only straightedge and compass.
Important note: If we simply use GeoGebra's built-in tool to construct regular n-gons, that would be cheating because there's no "n-gon Tool Postulate" in Euclid's system! We must use only the five postulates.
The Construction and Proof
Interactive Construction
Follow along with the step-by-step construction below
QED - We have successfully constructed an equilateral triangle using only Euclid's postulates!
A Global Overview of Euclid's Elements
Now that we've seen Euclid's very first proposition, let's step back and appreciate the magnificent architecture of his entire work. The Elements consists of thirteen books, and like any great piece of literature or music, it has a carefully planned structure with dramatic high points and a stunning finale.
The Grand Design
Euclid didn't just randomly collect geometric facts. Every proposition is strategically placed, building the tools needed for increasingly sophisticated results. It's like watching a master craftsman who starts with simple tools and gradually builds up to creating masterpieces.
Book I: The Journey from Triangle to Square
Book I tells a remarkable story. It begins with the humblest possible construction—making a triangle from two points—and culminates with one of the most famous theorems in all of mathematics.
The opening move: Proposition I.1 constructs an equilateral triangle. We start with just two points and create our first geometric object.
The crucial stepping stone: By Proposition I.46, Euclid has built up enough machinery to tackle a much more sophisticated challenge—constructing a square on a given line segment.
Proposition I.46
Statement: To describe a square on a given straight line.
This might seem like a simple task, but think about it: Euclid must prove that his construction actually creates four equal sides and four right angles, using only the postulates and previously proven propositions!
But why does Euclid need to construct squares? Because he's building toward his masterpiece...
The Pythagorean Theorem: Euclid's Masterpiece
Book I culminates with Proposition I.47—Euclid's version of the Pythagorean theorem. But here's what's fascinating: Euclid's statement is not the algebraic formula \(a^2 + b^2 = c^2\) that we learn in school!
Proposition I.47: The Pythagorean Theorem
Euclid's statement: "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle."
In modern terms: If we have a right triangle with legs of lengths \(a\) and \(b\), and hypotenuse of length \(c\), then the area of the square built on the hypotenuse equals the sum of the areas of the squares built on the two legs.
Mathematical statement: \(\text{Area}(C) = \text{Area}(A) + \text{Area}(B)\)
where \(A\), \(B\), and \(C\) are the actual geometric squares constructed on the sides of the triangle.
Euclid's "Windmill" Proof of the Pythagorean Theorem
The areas of the smaller squares add to the larger one.
This is why Proposition I.46 was essential! Euclid needed to prove he could construct squares before he could talk about their areas. For Euclid, the Pythagorean theorem is fundamentally a linear relation in areas, not a quadratic relationship in lengths!
The beauty doesn't stop there. Immediately following his proof, Euclid gives us:
Proposition I.48: The Converse
In other words: if the area relationship holds, then the triangle must have a right angle.
With these two propositions, Euclid completely characterizes right triangles in terms of the areas of squares on their sides.
The Elements' Grand Finale
If Book I ends with the Pythagorean theorem, what could possibly serve as the finale for all thirteen books of the Elements?
Euclid saves his most spectacular result for last. The final propositions of Book XIII accomplish something that seems almost impossible: they provide a complete classification of the regular solids—three-dimensional shapes with perfect symmetry.
The Final Achievement
Book XIII culminates with the construction of all five Platonic solids and—crucially—the proof that these five are the only ones possible.
Think about this: Euclid doesn't just show how to build these beautiful objects; he proves that his list is complete. No matter how creative you are, how hard you try, and how long you wait, nobody, not even an advanced alien civilization, will ever find a sixth regular solid!
This is mathematics at its most profound. Starting from five simple postulates about drawing lines and circles, Euclid builds up through hundreds of propositions to arrive at a definitive answer to a deep question about the nature of three-dimensional space.
So what exactly are these mysterious Platonic solids that provide such a fitting finale to the Elements?
What Are Platonic Solids?
The Platonic solids represent the ultimate expression of geometric perfection in three dimensions. Just as regular polygons (equilateral triangles, squares, regular pentagons, etc.) represent perfect symmetry in the plane, the Platonic solids represent perfect symmetry in space.
Definition: Platonic Solid (Regular Polyhedron)
A Platonic solid is a three-dimensional polyhedron that satisfies three conditions of perfect regularity:
- All faces are congruent regular polygons: Every face is exactly the same shape and size
- All edges have the same length: Perfect uniformity in every dimension
- All vertices are identical: The same number of faces meet at each vertex, at the same angles
These conditions ensure that the solid "looks the same" from every face, every edge, and every vertex.
The natural question—and the one that provides such a satisfying conclusion to the Elements—is: How many such perfect solids exist?
Unlike the infinite family of regular polygons (triangle, square, pentagon, hexagon, ...), the answer for three-dimensional solids is finite and small. Let's discover why by examining each possibility systematically.
The key insight is that we need to consider how many regular polygons can meet at a single vertex without causing the surface to fold in on itself or become flat.
Cases with Triangular Faces
We start by considering how many equilateral triangles can meet at a single vertex.
Case 1: Three Equilateral Triangles at Each Vertex
Result: Tetrahedron (4 faces)
When exactly three equilateral triangles meet at each vertex, we get the most basic of the Platonic solids.
Interactive Tetrahedron
Click and drag to rotate • The tetrahedron: 4 triangular faces, 6 edges, 4 vertices
Case 2: Four Equilateral Triangles at Each Vertex
Result: Octahedron (8 faces)
Think of this as two square pyramids glued together at their bases, but with triangular faces.
Interactive Octahedron
Click and drag to rotate • The octahedron: 8 triangular faces, 12 edges, 6 vertices
Case 3: Five Equilateral Triangles at Each Vertex
Result: Icosahedron (20 faces)
This is the most complex of the triangle-based Platonic solids.
Interactive Icosahedron
Click and drag to rotate • The icosahedron: 20 triangular faces, 30 edges, 12 vertices
Case 4: Six Equilateral Triangles at Each Vertex
Result: Impossible as a 3D solid!
Each equilateral triangle has 60° angles at its vertices. Six triangles meeting at a point would create 6 × 60° = 360°, which lies completely flat and creates a tiling of the plane that goes on forever, not a finite three-dimensional solid.
Interactive Triangular Tiling
Click and drag to rotate • Six triangles around each vertex create a flat tiling
Case 5: Seven or More Triangles at Each Vertex
Seven triangles would create 7 × 60° = 420° > 360° around each vertex, meaning that the triangles would have to get all "crumpled" up:
Interactive: 7 equilateral triangles at a point
Click and drag to rotate • Seven triangles meeting at a vertex create negative curvature
Hyperbolic Geometry in Physical Form
Click to view: A crocheted model of the hyperbolic plane by mathematician Daina Taimina. The characteristic "ruffled" surface shows how excess curvature manifests in physical space when more than six triangles meet at each vertex.
A Brief Aside: Non-Euclidean Geometry
Fast Forward 2000 Years...
For over two millennia, mathematicians were deeply bothered by Euclid's fifth postulate. Unlike the first four postulates, it was much longer and didn't seem "self-evident." It felt more like a theorem—a fact that should be proven from more basic principles—rather than a fundamental assumption.
In the 19th century, brilliant mathematicians including Gauss, Lobachevsky, Bolyai, Riemann, and Poincaré tried a radical approach: they assumed that the fifth postulate was false and attempted to derive a contradiction.
Their strategy was to prove a cascade of increasingly bizarre consequences, hoping that eventually they would arrive at a logical contradiction that would prove the parallel postulate must be true.
Plot twist: They never found a definitive contradiction! Instead, they discovered new alternative geometries with some very strange properties.
Example from Hyperbolic Geometry
Assumption: The angles in a triangle always add up to less than 180 degrees.
Surprising consequence: In this geometry, any pair of triangles with the same three angles must be congruent (exactly the same size and shape)!
This is completely different from Euclidean geometry, where triangles can have the same angles but different sizes (similar but not congruent triangles).
The discovery of non-Euclidean geometries was one of the most revolutionary developments in mathematics, showing that Euclid's geometry, while useful, is not the only possible way to understand space.
This connects back to our hyperbolic triangle example above: when more than six triangles meet at a vertex, we're no longer in ordinary flat space but in the curved world of hyperbolic geometry.
To understand this topic more deeply, watch this Veritasium video:
Conclusion for triangles in ordinary 3D space: We get exactly three Platonic solids using triangular faces.
Now let's return to our main story: the classification of perfect solids in Euclidean space.
Cases with Square Faces
Case 1: Three Squares at Each Vertex
Result: Cube (6 faces)
Each square has 90° angles, so three squares meeting at a vertex create 3 × 90° = 270° < 360°, allowing for a proper three-dimensional corner.
Interactive Cube
Click and drag to rotate • The cube: 6 square faces, 12 edges, 8 vertices
Case 2: Four Squares at Each Vertex
Result: Impossible as a 3D solid!
Four squares would create 4 × 90° = 360°, which lies flat and tiles the plane (like a checkerboard pattern).
Interactive Square Tiling
Click and drag to rotate • Four squares around each vertex create a flat tiling
And 5 or more squares could only tile the hyperbolic plane, same as triangles.
Conclusion for squares: We get exactly one Platonic solid using square faces.
Cases with Pentagon Faces
To understand pentagons, we need to calculate the interior angle of a regular pentagon.
Pentagon Division into Triangles
A pentagon can be divided into 3 triangles from any vertex
As shown above, a pentagon can be divided into 3 triangles, so its total angle sum is 3 × 180° = 540°. Since all five angles are equal, a regular pentagon has: 540° ÷ 5 = 108° per angle.
Case 1: Three Pentagons at Each Vertex
Result: Dodecahedron (12 faces)
Three pentagons create 3 × 108° = 324° < 360°, allowing for a proper three-dimensional corner.
Interactive Dodecahedron
Click and drag to rotate • The dodecahedron: 12 pentagonal faces, 30 edges, 20 vertices
Case 2: Four or More Pentagons at Each Vertex
Result: Impossible!
Four pentagons would create 4 × 108° = 432° > 360°, which again doesn't close up.
Conclusion for pentagons: We get exactly one Platonic solid using pentagonal faces.
Cases with Hexagon Faces and Beyond
A regular hexagon can be divided into 4 triangles, so its angle sum is 4 × 180° = 720°. Each angle measures 720° ÷ 6 = 120°.
Alternatively, we can think of a hexagon as composed of 6 equilateral triangles meeting at the center, with two triangles meeting at each vertex of the hexagon.
Case 1: Three Hexagons at Each Vertex
Result: Impossible as a 3D solid!
Three hexagons would create 3 × 120° = 360°, which lies completely flat and tiles the plane (like a honeycomb pattern).
Interactive Hexagonal Tiling
Click and drag to rotate • Three hexagons around each vertex create a flat honeycomb tiling
Case 2: Heptagons and Beyond
Result: All impossible in ordinary 3D space!
For any regular \(n\)-gon with \(n \geq 7\), the interior angles are too large. Even three such polygons meeting at a vertex would exceed 360°, which cannot exist in flat three-dimensional space.
For example, a regular heptagon (7-sided polygon) has interior angles of \(\frac{(7-2) \times 180°}{7} = \frac{900°}{7} \approx 128.6°\). Just three such angles would give us about 385.8° > 360°.
Conclusion for hexagons and beyond: No Platonic solids can be formed using hexagons or any larger regular polygons.
The Complete Classification
Let's summarize our systematic investigation:
The Five Platonic Solids
Using Triangular Faces:
- Tetrahedron: 3 triangles at each vertex → 4 faces
- Octahedron: 4 triangles at each vertex → 8 faces
- Icosahedron: 5 triangles at each vertex → 20 faces
Using Square Faces:
- Cube: 3 squares at each vertex → 6 faces
Using Pentagonal Faces:
- Dodecahedron: 3 pentagons at each vertex → 12 faces
The Remarkable Conclusion
These five are the only regular polyhedra that can exist in three-dimensional space.
This is one of the most beautiful classification theorems in mathematics. Unlike regular polygons (which form an infinite family), regular polyhedra form a finite family of exactly five members.
No matter how creative or persistent you are, you cannot discover a sixth Platonic solid. The mathematics simply won't allow it.
This classification was known to the ancient Greeks and provides a fitting culmination to Euclid's Elements. Starting from five simple postulates about points, lines, and circles, we can build up enough mathematical machinery to completely solve this deep question about the nature of three-dimensional space.
The proof that these five solids exhaust all possibilities demonstrates the power of systematic mathematical reasoning. By carefully analyzing the constraints imposed by geometry—specifically, the requirement that the angles around each vertex must sum to less than 360°—we can definitively answer a question that might seem impossibly complex.
Homework: Prove the theorem that if the angles in a triangle add to less than 180°, then AAA (angle-angle-angle) implies not just similarity (as in Euclidean geometry), but congruence! That is, given two triangles with the same corresponding angle measures, the corresponding side lengths must also be the same! (How can this be right? Surely there's a contradiction here somewhere?...)