Heptagon Seven Pointed Star Geometry

A heptagon is a seven-sided polygon, a closed two-dimensional shape with seven straight sides, seven vertices, and seven angles. The name comes from the Greek words “hepta,” meaning seven, and “gonia,” meaning angle.

The math on the interconnecting shapes shown in Figure 1 are fascinating, and we’ll be exploring that more in this article. We can also fold these shapes similar to how we did the Chestahedron and the Star Fort Crystal. We created 3D models to show what these new folded shapes look like, and we’re previewing those below in section 8.

In Figure 2, we’re showing the different relationships contained within a heptagon — which is indicated by the outer red outline. Connecting the vertices of the heptagon will create two alternative seven pointed stars, depending on which vertices are being connected.

These lines are shown in blue for the minor star points, and the green lines define the major star points. So we have $a = red$, $\; b = blue$, $\;$and $\; c = green$ line segments to keep in mind from this illustration as we work out the math.

1

Heptagon Geometry: Exploring 7-Pointed Stars

The heptagonal etymology reflects the ancient study of shapes and their properties, which continues to be a vital part of modern mathematics. As with any polygon, the heptagon’s sides and angles are connected in a closed chain, forming a two-dimensional shape. This closed chain can be imagined as a loop, each segment contributing to the overall form and symmetry that characterizes the heptagon.

Types of Heptagons

Heptagons can be categorized into two main types: regular and irregular. Each type brings its own set of characteristics and visual appeal.

• Regular Heptagon: All sides and angles of a regular heptagon are equal. This symmetry makes it a particularly appealing shape, often used in various designs and architectural features. The regular heptagon’s uniformity is not just aesthetically pleasing; it also makes it a favorite for mathematical exploration, illustrating concepts such as symmetry and balance.
• Irregular Heptagon: An irregular heptagon has sides and angles that are not equal. It can take on many different forms, depending on the lengths of its sides and the measures of its angles. This variability allows for a wide range of creative expressions in design and art. Each irregular heptagon tells a different story, reflecting the diversity and complexity of geometric forms.

2

The Geometry of a Heptagon

Heptagon Angles

Understanding the angles of a heptagon is crucial to comprehending its geometry. The sum of the interior angles of any polygon can be calculated using the formula:

Sum of interior angles = $(n – 2) \times 180^\circ$

where $n$ is the number of sides. For a heptagon:

Sum of interior angles = $(7 – 2) \times 180^\circ = 900^\circ$

Each interior angle of a regular heptagon is equal, and you can calculate it by dividing the total sum by the number of angles:

Each interior angle = $\frac{900^\circ}{7} \approx 128.57^\circ$

An example of an interior angle in Figure 3 is indicated by $\angle JHF$.

This precise calculation of angles is not only a mathematical exercise but also a foundational element in constructing geometrically accurate shapes. The understanding of these angles aids in the creation of various designs and structures that utilize the heptagon’s unique properties.

Exterior Angles

The exterior angles of a polygon are equally important. The sum of the exterior angles of any polygon is always $360^\circ$. For a regular heptagon, each exterior angle is:

Each exterior angle = $\frac{360^\circ}{7} \approx 51.43^\circ$

An example of an exterior angle in Figure 3 is indicated by $\angle DFE$.

These angles are instrumental when constructing a seven-pointed star from a heptagon. The exterior angles provide the framework for extending the heptagon’s sides into a star, showcasing the versatility of this shape. This concept is not only practical but also enhances the aesthetic appeal of designs incorporating heptagons.

Radial Internal Angles

We can also see that the radial lines emanating from the origin to the heptagon’s vertices form angles which are are calculated the same way as the exterior angles.

Each radial internal angle = $\frac{360^\circ}{7} \approx 51.43^\circ$

An example of a radial internal angle in Figure 3 is indicated by $\angle DOF$.

So the radial internal angles are equal to the external angles. The radial internal angles also define the apex of the “pie slice” triangles inside of the heptagon, like triangle $\bigtriangleup DOF$ for example. These are important triangles to consider in our research, but not to be confused with a “heptagonal triangle” which we’ll be discussing below in section 3. They don’t seem to have a dedicated name that we can find, so we’ll call them “Radial Heptagonal Triangles.”

Golden Ratio Relationship

In Figure 3, you’ll notice that we included the same two golden ratio proportioned circles that we used in our Star Fort Crystal Geometry article. The inner red circle has a radius $R_1$ equal to $\varphi$, and the outer red circle has a radius $R_2$ equal to $\varphi^2$. This relationship will give us an alternative way to make calculations as we explore the math further. In addition, it’s always nice to see where the golden relationships are present in a shape when they exist! Since we know this relationship does exist however, and we know the exact value of the golden ratio, we’ll use that as our baseline for further evaluation.

Radial Heptagonal Triangles

We can look at triangle $\bigtriangleup DOF$ as an example of the seven Radial Heptagonal Triangles that make up the inside of a heptagon in Figure 3. Since we’re using the golden ratio measurements of $R_1$ and $R_2 \;$ as our baseline for calculations, we can see that the height (or apothem) of triangle $\bigtriangleup DOF$ is equal to $\varphi$.

Since this triangle is isosceles, we know the two base angles are equal. We can also see that these base angles will be equal to half of the internal angle calculated earlier. If we let $\alpha$ be one of these base angles and let $\lambda$ be the apex angle, we can define these angles as follows,

\begin{align}
\alpha = \frac{450^\circ}{7} \\\\
\lambda = \frac{360^\circ}{7} \\
\end{align}

Also notice that our heptagon is inscribed within a blue circle with radius $R_3$, and we have enough information to calculate that now,

\begin{align}
R_3 = \frac{R_1}{\sin \left(\alpha \right)} = \frac{\varphi}{\sin \left(\alpha \right)} = 1.79588224 \\
\end{align}

We can also see that $R_3$ is the side length of the radial heptagonal triangles.

Minor Star Points

Two types of stars can be formed from a heptagon as you can see in Figures 1 and 2. But in Figure 3, triangle $\bigtriangleup BCD$ represents one of the minor star points. Since we know $R_1$ and $R_2 \; $ are in the golden proportions, we can calculate the height $h$ or apothem of triangle $\bigtriangleup BCD$,

\begin{align}
h = R_2 \; – \; R_1 = \varphi^2 \; – \; \varphi = 1 \\\\
\end{align}

Since we can see how the minor star point triangles are isosceles, their base angles $\lambda$ will be equal. So we can easily calculate the apex angle $\beta$ of our minor star point $\bigtriangleup BCD$,

\begin{align}
\beta = \angle BCD = 180^\circ \; – \; 2\lambda = \frac{540^\circ}{7} \\\\
\end{align}

Along with the side length $BC$ of our minor star point, it would be a good time to also calculate the edge length $BD$ of the heptagon in Figure 3 — according to our golden ratio baseline.

\begin{align}
\sin \left(\lambda \right) = \frac{h}{BC} =\frac{1}{BC} \\\\
BC = \frac{1}{\sin \left(\lambda \right)} \\
\end{align}

Continual Heptagon Replication

The interesting thing about a heptagon is you have continual replication of larger and smaller heptagons being generated in both directions. This is similar to how a five pointed star can be continually generated within a single pentagon. We should name each of these four heptagons in our illustrations to avoid confusion. So starting with the first outermost to the fourth innermost heptagons, we can define these as the primary, secondary, tertiary, and quaternary heptagons respectively.

In our other heptagon illustrations, we’ll be referring to edge length $AC$ of the outer (or primary) heptagon shown in Figure 2 as the variable $a$. We’ll solve for $a$ using our golden ratio proportions in section 3 below. But for now we can solve for $BD$ in Figure 3, which is the edge length of the inner (or secondary) heptagon defined by vertices $BDFHJLN$,

\begin{align}
\cos \left(\lambda \right) = \frac{BD}{2} \cdot \frac{1}{BC} \\\\
2BC\cos \left(\lambda \right) = BD \\\\
BD = \frac{2\cos \left(\lambda \right)}{\sin \left(\lambda \right)} = 2\cot \left(\lambda \right)
\end{align}

Secondary Heptagon Area

Since we have a value for the base length $BD$ and height $R_1$ of a radial heptagonal triangle, we can calculate the area $A_{rht}$ of these triangles,

\begin{align}
A_{rht} = \frac{1}{2}BD \; R_1 = \varphi \cot \left(\lambda \right) \\
\end{align}

Since seven of these triangles fill the area $A_{H2}$ of our Secondary Heptagon in Figure 3, we can easily calculate that area now,

\begin{align}
A_{H2} = 7A_{rht} = 7 \varphi \cot \left(\lambda \right) \\
\end{align}

Keep in mind that we’re using golden ratio proportions as our baseline for calculations. Even if we choose a different baseline however, the golden proportions are still inherent within the shape. So we can’t go wrong using inherent properties as a baseline, and we think golden proportions are are the best choice when that relationship exists. So part of our exploration in this article is to see if the golden ratio baseline will reveal any additional surprises!

3

Heptagon Diagonal Calculations

Now that we have most of the angles and measurements we need, we can move on to calculating the values for $a$, $b$, and $c$ that we illustrated earlier in Figure 2. Once we have some of the key heptagon diagonal angles, we can easily calculate aan exact value of $a$ based on $\varphi$. So this will make it easier to calculate $b$ and $c \;$ if we want to maintain a consistent model with the golden heptagon. But as we’ll see, our calculations of $b$ and $c$ don’t need to be dependent on a specific value of $a$, and we can choose any side length for the heptagon that we might want.

Heptagon Diagonal Angles

We can also get more of the angles created by the diagonals $b$ and $c$ in Figure 4. From our Basics of Circle Theorems article, we can use our theorem 3 which states that the angle subtended at the center of a circle by an arc is twice the angle subtended at any point on the circumference. In simpler terms, if you draw two radii to form an angle at the center and another angle at the circumference from the same arc, the angle at the center will always be twice that at the circumference.

From this we can see angles $\angle DGC$ and $\angle CGB$ (denoted by $\theta$ in Figure 4) will be equal to half of our central radial angle $\lambda$.

\begin{align}
\theta = \frac{\lambda }{2} = \frac{180^\circ}{7} \\
\end{align}

Furthermore, since line segments $AB$ and $CG$ are parallel, we can see that angle $\angle ABG$ is also equal to $\theta$. We can double check this by looking at half of triangle $\bigtriangleup ABG$ and solving for angle $\angle ABG$ by subtracting the two known angles $90^\circ$ and $\alpha$ from $180^\circ$,

\begin{align}
\angle ABG = 180^\circ \; – \; \left(\alpha + 90^\circ\right) = 90^\circ \; – \; \alpha \\\\
\angle ABG = 90^\circ \; – \; \frac{450^\circ}{7} = \frac{180^\circ}{7} = \theta \\
\end{align}

Heptagon Diagonal Measurements

Since we calculated the height $h = 1$ of a minor star point in Figure 3 and we now have a value for $\theta$, we can see that the edge length $AB = a$ of our outer primary heptagon would be the following,

\begin{align}
a = \frac{1}{\sin \left(\theta \right)} = 2.304764871
\end{align}

This gives us an exact value for $a$ based on the golden proportions, but we also have enough information about the heptagon by now to calculate $b$ and $c$ in any number of different ways. One fairly simple way to get $b$ is to look at the right triangle formed by splitting triangle $\bigtriangleup GAB$ in half and solving for $b$ with trigonometry,

\begin{align}
a\cos \left(\theta \right) = \frac{b}{2} \\\\
b = 2 \; a\cos \left(\theta \right) = 2 \; a\cos \left(\frac{180^\circ}{7}\right) = 1.802 \; a \\
\end{align}

Similarly, by dividing triangle $\bigtriangleup DGC$ in half to form two right triangles (as shown by the dashed line $GH$ in Figure 4), we can solve for $c$ related to half of $a$,

\begin{align}
c \; \sin \left(\frac{\theta }{2}\right) = \frac{a}{2} \\\\
c = \frac{a}{2 \; \sin \left(\frac{90^\circ}{7}\right)} = 2.247 \; a \\
\end{align}

We calculated $a$ using our golden proportions in the formula above, but these two equations for $b$ and $c$ will allow us to use whatever value of $a$ we want. If we let the side length of our heptagon $a = 1 \;$ for example, then $b = 1.802 \;$ and $c = 2.247$. But sticking to our golden ratio model, we’ll go ahead and grab an exact value of $b$ which we can use for further calculations on the Inner Heptagon Geometry in Section 5.

\begin{align}
b = 1.802 \; a = 4.153186298 \\
\end{align}

Primary Heptagon Area & Perimeter

In Section 2 above, we calculated the area of our secondary heptagon $A_{H2}$, and we now have enough information to calculate $A_{H1}$. We can evaluate one of the primary radial heptagonal triangles first, so we’ll find the area $A_{COD}$ of triangle $\bigtriangleup COD$. We already have the angles we need, and we know that the side length of these trianlges is $R_2$. So we need to find the height $OH$ first.

\begin{align}
OH = \frac{R_2}{\sin \left(\alpha \right)} = \frac{\varphi^2}{\sin \left(\alpha \right)} \\\\
A_{COD} = \frac{OH}{2}a \; = \frac{\varphi^2\sin \left(\alpha \right)}{2\sin \left(\theta \right)} \\
\end{align}

Multiplying the area of one of these triangles by seven, we can calculate the final area of our primary heptagon $A_{H1}$,

\begin{align}
A_{H1} = \frac{7\varphi^2\sin \left(\alpha \right)}{2\sin \left(\theta \right)} = 19.02741259 \\\\
\end{align}

We should also include the perimeter of the primary heptagon $P_{H1}$, which would simply be seven times the side length $a$,

\begin{align}
P_{H1} = 7a = \frac{7}{\sin \left(\theta \right)} = 16.1333541 \\
\end{align}

4

Heptagon Math And The Optics Equations

Using Ptolemy’s Theorem For Heptagon Equations

We think the best way to explore the math in a heptagon is to make use of Ptolemy’s Theorem. This is because the heptagonal shape lends itself to evaluating the diagonals of several cyclic quadrilaterals as we’re showing in Figures 5 and 6.

We already know that a circle can be circumscribed to meet up perfectly with all exterior vertices of a regular heptagon. So we can safely show these illustrations without the circles, and it gives us another interesting way to visualize Ptolemy’s Theorem.

If we let $a$, $b$, and $c$ represent the red, blue, and green lines respectively, we can work our way from left to right in Figure 5 and write three new equations for $a^2$, $b^2$, and $c^2$.

\begin{align}
a^2 = c\left(c \; – \; b\right) \\
b^2 = a\left(c + a\right) \\
c^2 = b\left(a + b\right) \\
\end{align}

The Heptagonal Triangle

A heptagonal triangle is an obtuse, scalene triangle whose vertices coincide with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex). Thus its sides coincide with one side along with the adjacent shorter and longer diagonals of a regular heptagon. All heptagonal triangles are similar (have the same shape), and so they are collectively known as “the heptagonal triangle.”

The Optics Equation

In figure 6, a heptagonal triangle is defined by the vertices $\bigtriangleup ABC$, and it has side lengths $a$, $b$, and $c$. So these same side lengths will satisfy the three equations we derived earlier from Ptolemy’s Theorem. An interesting aspect of these equations is that they can reveal another important formula known as “the optics equation” through a series of algebraic manipulations. But a much easier way to see this relationship is to divide both sides of our Ptolemy’s Theorem derivation in Figure 6 by the lowest common denominator. In this case, that would mean dividing everything throughout by $abc$.

\begin{align}
\frac{ab}{abc} + \frac{ac}{abc} = \frac{bc}{abc} \\\\
\end{align}

Cancelling like terms and rearranging our formula, we can see how the optics equation is indeed revealed within the heptagon,

\begin{align}
\frac{1}{a} = \frac{1}{b} + \frac{1}{c} \\
\end{align}

The optics equation is a very significant relationship to notice, since it appears in numerous contexts of geometry. The Inverse Pythagorean Theorem and the Crossed Ladders Problem are two important examples. But it also appears in The Mirror Equation, The Thin Lens Equation, Electrical Engineering, Paper Folding, and The Harmonic Mean. And if that’s not more than enough, it’s also related to Fermat’s Last Theorem!

Geometric Optics Lesson

We can see the importance of optics equations and we know they have a direct relationship to the heptagon. So we should pay closer attention to this formula as we’re studying heptagons and heptagrams (or seven pointed star geometry). The video below explains the mirror equation and the thin lens equation in more detail, and this lesson also has a part 2 if you want to learn more.

5

Inner Heptagon Geometry

In Figure 7, we can begin to see the continuous replication of a heptagon, extending forever into larger and smaller sizes. In the illustration for example, we can see four heptagons outlined in red. We know the outer two blue circles are defined by radii $R_1$ and $R_2$ which are equal to $\varphi$ and $\varphi^2$ squared respectively.

Since we know the outer two circles are related by golden proportions, we know the inner two white circles will be as well. Also notice that each of the two inscribed hexagons within these circle pairs are rotated by a cycle of $\frac{\alpha }{2} = \frac{180^\circ}{7}$ degrees out from each other.

Heptagon Pair Scaling Factors

What would be interesting to see is what the scaling factor is between this inner and outer pair of hexagons. In other words, how much would you scale down the outer pair to produce the inner pair?

Again referring to Figure 7, we need to solve for $r_1$, $r_2$, and $r_3$. Due to the symmetry of a heptagon, we can see that diagonals $AD$ and $CG$ are parallel to sides $BC$ and $AB$ respectively. Accordingly, parallelogram $ABCH$ is a rhombus with sides equal to the heptagon’s edge length, $a$.

Since we know that diagonal $AC$ is equal to the value $b$ we calculated earlier, we can find length $BH$.

\begin{align}
a^2 = \left(\frac{b}{2}\right)^2 + \left(\frac{BH}{2}\right)^2 \\\\
BH = \sqrt{4a^2 \; – \; b^2} = 1.999701982 \\
\end{align}

From here, we can find a value for $r_3$,

\begin{align}
r_3 = \varphi^2 \; – \; BH = 0.618332008 \\
\end{align}

Notice that $r_3$ is the side length of our innermost radial heptagonal triangles, so we can calculate $r_1$ using a similar formula to what we developed in Section 2,

\begin{align}
r_1 = r_3\sin \left(\alpha \right) = 0.5570978892 \\
\end{align}

Innermost Heptagon Minor Star Points

Next we’ll want to also calculate a value for $r_2$, but to do that we need to evaluate the minor star points of our innermost yellow heptagram. In Figure 8, we’ve isolated one of these star points (outlined in black), and we need to solve for base $d$, height $h_2$, and side length $s$. We can start by finding $d$ with the Pythagorean Theorem,

\begin{align}
\left(\frac{d}{2}\right)^2 + r_1^2 = r_3^2 \\\\
d = 2 \; \sqrt{r_3^2 \; – \; r_1^2} = 0.5365684074 \\
\end{align}

Earlier in Section 2, we solved for the minor star point $\bigtriangleup BCD$ in Figure 3. This triangle is similar to our innermost minor star point in Figure 7, so we can find $s$ with the following ratio equality,

\begin{align}
\frac{BD}{BC} = \frac{d}{s} \\\\
s = d \frac{BC}{BD} = 0.4302944537 \\
\end{align}

Now we can get $h_2$, and finally solve for $r_2$,

\begin{align}
h_2 = s\sin \left(\lambda \right) = 0.3364177506 \\\\
r_2 = h_2 + r_1 = 0.8935156398 \\
\end{align}

Scaling Factors of Replicating Pairs

Our theory is, everything contained within the circle defined by $r_2$ will be uniformly scaled up if we find the scaling factor from $r_2$ to $R_2$. So we can let $x$ be the amount multiplied by $r_2$ to reach $R_2$,

\begin{align}
r_2\left(x\right) = R_2 \\\\
x = \frac{R_2}{r_2} = \frac{\varphi^2}{r_2} = 2.930037117 \\
\end{align}

We can also scale the outer heptagon pair down as follows,

\begin{align}
R_2\left(x\right) = r_2 \\\\
x = \frac{r_2}{R_2} = \frac{r_2}{\varphi^2} = 0.3412926048 \\
\end{align}

Circumferences of Heptagon Circles

We now have the radii for six different circles relating to our heptagon, so it’s straight forward to calculate the circumference for each of these. We’ll define each circle by its radius and provide numbers for each circumference,

\begin{align}
C_{R_1} = 2\pi R_1 = 10.16640739 \\\\
C_{R_2} = 2\pi R_2 = 16.4495927 \\\\
C_{R_3} = 2\pi R_3 = 11.2838609 \\\\
C_{r_1} = 2\pi r_1 = 3.500349272 \\\\
C_{r_2} = 2\pi r_2 = 5.61412434 \\\\
C_{r_3} = 2\pi r_3 = 3.885094588 \\
\end{align}

Areas of Heptagon Circles

We should also include the areas of these heptagon related circles, since this is another interesting set of numbers to have documented — in case we find a relationship with them somewhere else in our research,

\begin{align}
A_{R_1} = \pi R_1^2 = 8.224796348 \\\\
A_{R_2} = \pi R_2^2 = 21.5327964 \\\\
A_{R_3} = \pi R_3^2 = 10.1322427 \\\\
A_{r_1} = \pi r_1^2 = 0.9750185955 \\\\
A_{r_2} = \pi r_2^2 = 2.508153951 \\\\
A_{r_3} = \pi r_3^2 = 1.201139169 \\
\end{align}

Heptagonal Pi Derivation

Now that we have the areas and circumferences of these heptagonal circles — along with other key measurements, we can look for some potentially interesting relationships. For example, what happens when we combine these measurements, or subtract one measurement from another. One thing we can look at right away is what happens when we subtract the area of our primary heptagon $A_{H1}$ from the area of our outer circle $A_{R_2}$,

\begin{align}
A_{R_2} \; – \; A_{H1} = 21.5327964 \; – \; 19.02741259 = 2.50538381 \\
\end{align}

Looking through our list of heptagonal circle areas, we can see that this result is very close to the area of $A_{r_2}$. We used the standard version of $\pi$ to calculate $A_{R_2}$ and $A_{r_2}$ however, but what if we were to treat $\pi$ as an unknown in this special case, and then solve for it?

We know the standard version of $\pi$ is a very good estimation, but it can still only ever be an approximation. On the other hand, all of our other heptagonal numbers for the radii and the area of our primary heptagon $A_{H1}$ are based on exact number calculations. So we have everything we need to derive a heptagonal value of pi which we can call $\pi_h$,

We’ll start off by pretending that $A_{R_2} \; – \; A_{H1}$ is exactly equal to to $A_{r_2}$, and form the necessary equations as follows,

\begin{align}
\pi_h R^2_2 \; – \; A_{H1} = \pi_h r^2_2 \\\\
\pi_h\left(R^2_2 \; – \; r^2_2\right) = A_{H1} \\\\
\pi_h = \frac{A_{H1}}{R^2_2 \; – \; r^2_2} = 3.142050094 \\
\end{align}

As we can see, this is a fairly close approximation to the standard $\pi = 3.141592654$, and it’s even closer than the Archimedes approximation of $22/7 = 3.142857143$ that we derived in our Squaring the Circle Insights article.

Alternative Derivation of the Optics Equation

We should also notice in Figure 7 that triangles $\bigtriangleup AHC$ and $\bigtriangleup GHD$ are similar. This can easily be shown by applying the first theorem on our Basics of Circle Theorems page to the two chords $DC$ and $AG$. We won’t do the calculations here, but want to point out that using this similarity will also result in giving us the same optics equation we derived earlier with Ptolemy’s Theorem. So we have several ways to show how the optics equations are inherent within a heptagon.

6

Constructing a Seven-Pointed Star

A seven-pointed star, also known as a heptagram, can be derived from a heptagon. The heptagram is a complex, yet aesthetically pleasing shape that has been used in various cultural and artistic contexts. This star formation is more than just a decorative element; it is a testament to the creativity and ingenuity of geometric constructions.

Note that there are two different seven-pointed star construction types, and we’ll outline each one below. The first version has minor or shorter star points, and the second version has major or longer star points. You can see both of these star versions illustrated in Figures 1 and 2 above.

Steps to Draw Seven-Pointed Stars

1. Draw a Regular Heptagon: Start by constructing a regular heptagon, ensuring all sides and angles are equal. This foundational step is critical to achieving the desired symmetry in the final star shape.
2. (Minor Star Points) Connect Every Second Vertex: To form the star, connect every second vertex of the heptagon. This means you will skip one vertex each time you draw a line. This method creates a pattern of intersecting lines that enhance the star’s intricate design.
3. (Major Star Points) Connect Every Third Vertex: Referring to Figure 2 above, you’ll want to look for the opposite vertices and connect a line between either the left or right opposite vertex. If you choose the right opposite vertex (the third vertex to the right), your next line will connect back to the vertex immediately to the left of your starting point. Or if you chose the left opposite vertex for your first connecting line, you’ll choose the vertex immediately to the right of your starting point.
4. Complete the Star: Continue connecting the vertices until you return to the starting point. You should have a minor or major seven-pointed star with intersecting lines. This completed star not only represents mathematical precision but also offers a visually captivating image that has been utilized in various cultural symbols.

This star formation showcases how the heptagon’s geometry can be transformed into a more intricate and captivating design. The heptagram extends the heptagon’s properties, illustrating the dynamic possibilities of geometric shapes.

7

Practical Applications of Heptagons

Heptagons and their star counterparts are not just theoretical shapes; they have practical applications in various fields. These applications highlight the intersection of mathematics, art, and functionality, demonstrating the heptagon’s relevance in contemporary contexts.

Architecture and Design

Heptagons are often used in architecture and design to create visually appealing structures. Their symmetry and unique angles make them ideal for decorative elements, tiling patterns, and structural designs. Architects and designers leverage these properties to create innovative spaces that are both functional and aesthetically pleasing. The heptagon’s distinct shape offers endless possibilities for creative expression, from floor tiles to intricate facades.

Cultural Significance

The seven-pointed star holds cultural significance in many societies. For instance, it is used in the flags of Jordan and Australia, symbolizing unity and strength. In religious contexts, the heptagram is associated with mystical and spiritual meanings. This cultural resonance adds layers of meaning to the heptagon, enriching its role beyond mere geometry. The star’s presence in these symbols underscores its importance as a unifying and powerful emblem across different cultures.

Mathematics and Education

In mathematics education, heptagons are used to teach concepts such as symmetry, angles, and geometric construction. They provide a practical example of how mathematical principles are applied in real-world scenarios. Educators use heptagons to illustrate complex ideas in an accessible way, fostering a deeper understanding of geometry among students. The heptagon’s unique properties make it an excellent tool for engaging learners and demonstrating the beauty of mathematics.

Real World Examples

• Coins: The British 50-pence and 20-pence coins are seven-sided, though they have curved edges to allow them to roll smoothly in vending machines.
• Nature: Some flowers, like bitterweed, can have seven petals, and some natural formations or crystals may exhibit heptagonal patterns.
• Star Forts: In Figure 9, we’re showing a star fort design drawing for the Dutch city Coevorden, which is based on a heptagon.

8

Folded Star Crystals

In our Chestahedron and Star Fort Crystal articles, we demonstrated how to fold two types of five pointed stars to create new “star crystal” objects. The video below shows the process we used for folding a Secondary Five Pointed Star (an interesting shape we encountered when researching ancient star forts), and we used the same process to fold our Chestahedron Phi Model.

Star Fort Crystal Video

Heptagonal 3-Fold Star Crystal (Minor Star Points)

In a similar fashion to what we demonstrated above in the Star Fort Crystal video, we’re folding a heptagon to form a three sided heptoganal tetrahedron at the apex, where the side triangles in this case are the minor star points. We could call this first star formation with shorter star points the “Primary Heptagram.” Since these folded star shapes have never been defined before, we could call this one the “Primary Heptagram 3-Fold Crystal.”

Heptagonal 3-Fold Star Crystal (Major Star Points)

This animation shows the same process as above, but this time using the major star points of what we could call the “Secondary Heptagram.” So this new shape in the animation below could be called the “Secondary Heptagram 3-Fold Crystal.”

Heptagonal 5-Fold Star Crystal (Major Star Points)

This next animation increases the number of folded sides from three (as we have in the Star Fort Crystal and the Chestahedron) to five sides. Since we have more star points in a heptagram than we do in a pentagram, we have more folding options at the apex. What’s nice about this fold is that it gives us a pentagon at the base, which means it’s also related to a five pointed star! We’ll call this one the “Secondary Heptagram 5-Fold Crystal.”

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Relationship to Alchemy

The heptagon (a seven-sided polygon) or heptagram (a seven-pointed star) is significantly related to alchemy through its association with the seven known planets of antiquity and the seven alchemical substances (or elements).

Alchemical Symbolism

• Seven Planets: In ancient and medieval cosmology, only seven celestial bodies were visible to the naked eye: the Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn. These were central to alchemical theory and practice, as each planet corresponded to a specific metal and a certain stage in the alchemical process.
• Seven Substances/Elements: The heptagram, specifically, was used by alchemists to represent the seven primary alchemical substances or elements: fire, water, air, earth, sulfur, salt, and mercury.
• Alchemical Stages: The number seven is also connected to the seven primary stages of the magnum opus (the “Great Work”) of alchemy, which were believed to lead to the creation of the Philosopher’s Stone and the perfection of the human body and soul:
• Calcination (Calcinatio)
• Dissolution (Solutio)
• Coagulation (Coagulatio)
• Sublimation (Sublimatio)
• Mortification (Mortificatio)
• Separation (Separatio)
• Conjunction (Coniunctio)

Other Occult and Spiritual Meanings

Beyond alchemy, the heptagram (often called the Elven Star or Fairy Star) holds various other mystical meanings in different traditions, often representing:

• Divine perfection and harmony.
• A bridge between different realms or worlds.
• The seven days of Creation in Christian symbolism.
• The seven clans of the Cherokee people, symbolizing peace.

The heptagon appears in contemporary occult designs and is also referenced in popular culture, such as the Fullmetal Alchemist series, where it is used in transmutation circles, often for medicinal purposes or energy transfer.

Conclusion

The heptagon and its seven-pointed star geometry offer a rich exploration of mathematical beauty and practical utility. Whether you’re interested in the angles that define its shape, the method of constructing a heptagram, or its applications in various fields like geometric optics, the heptagon remains a fascinating subject of study. Its versatility and aesthetic appeal continue to inspire those who study and utilize its properties.

By understanding the geometry and significance of the heptagon, you can appreciate its role in both mathematics and everyday life. Whether used in design, education, or cultural symbolism, the heptagon’s seven-sided allure continues to captivate and inspire. Its enduring presence in diverse contexts is a testament to its importance and the timeless nature of geometric exploration.

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Cite This Page As:

The Math Zone. “Heptagon Seven Pointed Star Geometry.” From MathZone.io — A Modern Exploration of Ancient Mathematics. https://mathzone.io/heptagon-seven-pointed-star-geometry/

New Chestahedron Phi Model PDF Press Release
> A New Paradigm In Euclidean Geometry (PDF)

Work Sited:

Heptagon (2025) Wikipedia. Available at: https://en.wikipedia.org/wiki/Heptagon (Accessed: 22 October 2025).