Structures that exhibit octagonal symmetry are the Dome of the Rock and the soon to be built Freedom Tower in New York. The Freedom Tower will replace the Twin Towers of the World Trade Center complex. It's shape begins as a square at the base and rotates as it rises ending in another square at the top, which is offset from the base by 45-degrees. From above it would look like two overlapping squares or an octagon.
Facing the Twin Towers during 9/11 was the Colgate Clock CC, which is an octagonal structure. Colgate means "dark gate".
Parts of the Large Haydron Collider of CERN also exhibits octagonal symmetry. Even the Great Pyramid of Egypt is octagonal. It's sides are slightly concave towards the center, cutting each face into two. The website Octagon-Maat makes the claim that the layout for the Giza complex was based on octagons.
Octagonal symmetry is also found in the cyclical nature of the seasons on earth. There are the four points of the solstices and equinoxes, and there are also the four points between each of these, known as cross-quarter days. Each of these times of the year are marked by a holiday on what is known as the Yule Wheel. A similar wheel has been used to depict the four principal stages in the alchemical process plus the four operations of alchemy.
Octagonal symbols include the Buddhist Dharma Wheel which uses the 8-spoke wheel as a symbol of the eight-fold path of Buddhism. Similar symbols are seen in Mayan and Sumerian art. An 8-spoke wheel is found on the tomb of William Sinclair (St. Clair) at Rosslyn Chapel. This symbol is said to represent the Holy Grail. The 8-rayed star is a symbol for the goddess Ishtar/Innanna/Isis/Mary/Ashtarte and so on.
The Number 8
The number 8 turned on it's side is the symbol for infinity. It can be seen as two serpents twisting around eachother like the caduceus. The eighth letter of the english alphabet is H, and the letter is sometimes used as code for the number, such as in 88 for Heil Hitler. There are 88 modern constellations and 88 keynotes on a piano. The Freemasonic Checkerboard is a matrix of 8x8 (64) interlocking black and white squares. A pair of eights may be turned 90 degrees and strung together to form the four rings of the Audi car logo. The Greek name for Jesus (IHSOUS) equals 888 using letter-number values of Greek gematria.
The Egyptians believed that there were 9 realms (7 heavens and 2 lands/earths). We are on the 8th realm. The 8th realm is a replication of the First Principle, the image of God. 8 is an octave of 1. There are 7 notes on a piano, and the 8th note is an octave of the first. The octave is the future state of the past, the continuous creation in a series of replications. The renewal is not identical to the original, but is analogous to it.
- Coffin of Petamon text:
I am One who becomes Two,
who becomes Four,
who becomes Eight,
and then I am One again.
Eight is the number associated with Thoth. At Hermopolis, Thoth is called the Master of the City of Eight. The words of Ra which created the world were revealed through Thoth as soundwaves, which became all the creatures and things of this world. The number 8 is synonymous with creation and manifestation. At Memphis, Ptah in his 8 forms, created the universe. At Heliopolis, Atum created 8 divine beings. At Hermopolis, the Ogdoad or 8 primeval neteru, created the universe. And at Thebes, after Amen created himself in secret, he created the Ogdoad. (8 neteru) 
The eight pointed compass rose is composed of four cardinal directions (N - E - S - W) as well as four intermediary positions (NE-SE-SW-NW). Synchromusickally, this eight pointed star lines up with the common eighth-note subdivision of a whole-note duration. This common meter is written as 8/8.
The Chandogya Upanishad (1.1.1-1) states:
"The udgitha ["the chanting", aum, Om] is the best of all essences, the highest, deserving the highest place, the eighth."
"The one syllable [Aum] is indeed . This one syllable is the highest. Whosoever knows this one syllable obtains all that he desires.
A circle with 8 points marked along the edge, or 8-rays emanating from the center, will sometimes be equated with the Enneagram, a star or circle with nine-sided geometry. The reason for this is that the central point of the octagon will be counted as a hidden central point; the hub of the wheel.
Octagons in popular culture
- In the movie Kung-Fu Panda, the main character is waking up to his full potential, fulfilling his destiny as the Dragon Warrior. He does it by obtaining the secret of the mysterious Dragon-scroll. The dragon-scroll is located high above an octagonal water door.
- In the movie Masters of the Universe a device called the "Cosmic Key" is used. It can open a portal to any location and any time. It is octagonal.
- In Minute Men the device used to travel in time is octagonal.
- The logo of the Alice project, which is part of the LHC, is an octagon.
- Every day a time ball is dropped from the Octagon Room in Greenwich. The Greenwich observatory where global time is defined.