My Geometry Notes

N-Dimensional Solids

Linking my 3D n-cubes to conventional geometry based on Coxeter's work. Coxeter has linked this geometry to conventional algebraic geometry. This was the break through to acceptance by algebraic graphing mathematicians.

Four Dimensions is Not so Difficult

There was Facebook science post about the old 2D ant walking around a plane in 3D. 2D paper to a 3D being is like 3D cube to a 4D being. A 3D being can wrap the paper's opposite sides together. It would look like a cylinder in 3-space. A 4D being can wrap the cube's opposite faces together. It would look like a cube doughnut in 4-space.

In 4-space, it's like making a torus using the shape of a square instead of a circle. Imagine a cube, grab the top in one hand the bottom in the other, then pull it wider like taffy, and turn the top to face the bottom. That's the equivlent of piece of paper rolled into a cylinder.

Or, use a sphere where to top point is p1, and the most bottom point is p2. In 4D, stretch sphere out into a cylinder. Then, turn cylinder into a torus where p1 meets p2.

N-Dimensional Solids

Dimensional solids,

Dimension	1	2	3	4	5	6	7	8	9	10	...
Number of faces	0	1	6	12	20	30	42	56	72	90	...

The formula for term index n, n starting at 2:
a_n = (n−1)n for n≥2.

whole list: 
0-cube  a point
1-cube  a line
2-cube  a square
3-cube  a cube
4-cube  a tesseract
5-cube  a penteract
6-cube  a hexeract
7-cube  a hepteract
8-cube  an octeract
9-cube  an enneract
10-cube a dekeract

5-cube See section: "The 5-cube is 5th in a series of hypercube: Petrie polygon orthographic projections ..." Hypercubes projected into two dimensions. Orthographic projection means representing three-dimensional objects in two dimensions. An orthographic projection map of the earth is like a photo taken far away from out in space, mathematically speaking, from an infinite distance which makes the projection lines parallel, you can see the top most and bottom most points. Any distance closer than infinite, the top most and bottom most points are not visible.

Orthographic projection of the earth.

For an n dimensional cube, there are 2ⁿ to the nth number of vertices.
For an n dimensional cube, there are n * (2^n-1) number of edges.
When when n is the first order of infinity, The number of vertices equals the number of edges, which equals the second order of infinity, the second transfinite number.
ℵ₀ = infinity such as the number of natural numbers.
2^ℵ₀ = next order of infinity such as the cardinality of the real numbers. Google Gemini AI note, Recent research into infinite-dimensional (and transfinite) polytopes has accelerated in the last two years (2023–2025), branching into Abstract Algebra, Quantum Physics, and Geometric Modeling.

Interesting

Creating a line segment in 1699 with parchment, ruler, ink, and quill pen.

Be cool to have the Deutsche Fotothek image(Geometry & Construction & Distance & Measuring Instrument) beside an image of me creating a line segment with laptop and LibreOffice Draw.

Article: An Open Logic Approach to EPM, with cube projection diagrams:

Geometric Definitions

Regular polygon is a polygon with equal length edges, equal angles, and convex . Examples, an equilateral triangle is a regular triangle which has equal sides and equal angles. A square is a regular quadrilateral. A rhombus quadrilateral is not a square, then it is not a regular quadrilateral because it's angles are not all equal even though it has equal length sides.
Convex polygon means that the line segment between two points of the polygon is contained in the boundary(inside) of the polygon. An example of a convex polygon: a regular pentagon.

In 1619 Kepler defined stellation for polygons as the process of extending edges until they meet to form a new polygon.
Stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. "Stella" in Latin, is "star". A compound to 2 squares where one is rotated 45^o from the other, creates a star like shape, a stellation polygon.
In Johannes Kepler's Harmonices Mundi (1619, The Harmony of the World), the rhombic dodecahedron is one of his "rhombic solids" which is a polyhedra with all faces being congruent rhombuses. The dodecahedron is composed of 12 faces and is related to the cube and octahedron

This PDF by Professor Sarah Hart of Gresham College, goes over the five Platonic solids are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Coxeter became a prominent mathematician, well-known for his work on geometry and symmetry, for example studying and classifying symmetries of higher dimensional figures. Coxeter and Roger Penrose met Escher for the first time, at an exhibition of Escher's work. and he bought a couple Penrose came up with ‘Penrose triangles’ after seeing Escher’s Relativity impossible staircase print. Infinity in a finite picture: hyperbolic tiling from Coxeter’s paper and Circle Limit I, M.C. Escher (1958).

Shapes

Notes, Thoughts

Where to begin? The popular response is, "at the beginning." Carl Sagan suggested a great beginning when he said, “If you want to make an apple pie from scratch, you must first create the universe.”

Lasting effects in Physics, Astronomy, Cosmology

Max Planck's Quanta (1900): Planck introduced the idea that energy is emitted and absorbed in discrete "bundles" called quanta (packets of energy).
Albert Einstein and the Photoelectric Effect (1905): Building on Planck's work, Einstein proposed that light itself consists of discrete particles (later called photons), which explained the photoelectric effect (where light striking a metal surface ejects electrons). This established the concept of wave-particle duality.
Ernest Rutherford's Atomic Nucleus (1911): Rutherford's experiments revealed the existence of a dense atomic nucleus, providing a qualitatively correct picture of the atom that was necessary for subsequent models.
Niels Bohr's Atomic Model (1913): Bohr applied quantum ideas to the hydrogen atom, proposing that electrons orbit the nucleus in specific, discrete energy levels or shells, and only "jump" between these allowed levels. His model successfully explained the spectral lines (colors) emitted by hydrogen
Relativity: Albert Einstein.
Werner Heisenberg's Matrix Mechanics (1925): Heisenberg developed the first formal mathematical framework for quantum mechanics, known as matrix mechanics.
Erwin Schrödinger's Wave Mechanics (1926): Based on Louis de Broglie's hypothesis that matter has wave-like properties, Schrödinger devised an alternative and more popular formulation: the Schrödinger equation. This equation describes the behavior of a quantum system's wave function (\(\Psi \)), which provides probability amplitudes for the outcomes of measurements.
Heisenberg's Uncertainty Principle (1927): Heisenberg formulated his famous uncertainty principle, which states that certain pairs of physical properties, such as a particle's position and momentum, cannot both be known with perfect accuracy simultaneously.
The Copenhagen Interpretation (Late 1920s): Niels Bohr, Heisenberg, and other physicists developed the Copenhagen interpretation, the first general consensus on how to understand the new theory's implications, emphasizing concepts like complementarity (wave-particle duality) and indeterminism.

Mathematics

Computer Hardware

1946, The first vacuum-tube computer, ENIAC: general-purpose, electronic computer was the ENIAC (Electronic Numerical Integrator and Computer), at the University of Pennsylvania
---- Quantum mechanics was absolutely essential for the development of silicon chips.
1945(or after), John Bardeen eventually developed a new branch of quantum mechanics known as surface physics. Bardeen and Walter Brattain eventually succeeded in building a working device.
1945 and after, Team of Twelve Bell Lab people developed the transistor.
1947, The first transistor was successfully demonstrated to colleagues and managers at Bell Labs on December 23, 1947, at Bell Laboratories.
1954, The first transistor computer, TRADIC: Bell Labs' machine for the U.S. Air Force, it was small (3 cubic feet) and used nearly 800 transistors, performing millions of operations per second.
1959, first practical monolithic silicon chip, forming the foundation for modern electronics by Robert Noyce (Fairchild Semiconductor, 1959).
1971, The first single-chip CPU was the Intel 4004, launched on November 15, 1971.
1974, first truly transparent resistive touch screen.
1975, Altair 8800 using Intel 8080 microprocessor: sparked the hobbyist computer revolution but wasn't a mainstream success.
1977, Apple II, TRS-80 (Tandy Radio Shack Z80), Commodore PET
1981, IBM PC which became the standard home and business small application computer architecture running Microsoft DOS.
1981, The first commercially successful portable computer was the Osborne 1, released in April 1981 by the Osborne Computer Corporation, known as the first "luggable"
1982, Commodore 64
1995, The first commercially available flat-screen TVs emerged in the mid-to-late 1990s, with Fujitsu introducing a 42-inch plasma TV in 1995,
1990s: The invention of efficient blue LEDs (Nobel Prize 2014) enables full-color LED displays, paving the way for modern screens.
2004: Sony releases the Qualia 005, the first LED-backlit LCD TV.
2007: Sony launches the XEL-1 OLED TV, the first commercially available OLED television.
2007, The first iPhone was announced by Steve Jobs on January 9, 2007, and officially released to the public in the United States on June 29, 2007, combining an iPod, phone, and internet communicator into one multi-touch device.
2008, The first Android operating system was Android 1.0, released publicly on September 23, 2008, with the first device being the HTC Dream (also known as the T-Mobile G1). This initial version included core Google apps like Gmail, Maps, Calendar, and YouTube, along with features like multitasking, GPS, and Bluetooth, and introduced the Android Market (now the Google Play Store).
2008, iPhone 3G which utilized Assisted GPS (A-GPS) for faster location fixes.

Computer Software

1843, The first computer algorithm, a step-by-step method to calculate Bernoulli numbers using Charles Babbage's theoretical Analytical Engine, was published in 1843 by Ada Lovelace.
Quantum mechanics was absolutely essential for the development of silicon chips.
late 1960s and early 1970s, creation of UNIX by Ken Thompson and Dennis Ritchie at Bell Labs.
early 1970s, creator of the C programming language: Dennis Ritchie. He developed the language at Bell Laboratories.
1970s, Creation of TCP/IP, the internet protocol by Vinton Cerf and Robert Kahn are credited as the creators of the TCP/IP
1975, Microsoft Operating System: Bill Gates and Paul Allen after they founded Microsoft.
1991, Creator of Linux: Linus Torvalds who began development.
1991, Creator of the Java programming language: primary creator and lead designer: Canadian computer scientist James Gosling.
1983, The Domain Name System (DNS) was invented by computer scientist Dr. Paul Mockapetris.
1989, Creator of HTML: Tim Berners-Lee
1990, Creators of web browsers: The first web browser and web server were created by Tim Berners-Lee in 1990, leading to the popular NCSA Mosaic created by Marc Andreessen and Eric Bina in 1993. Leading the to the Netscape web browser.
1994, Founders of amazon.com: Jeff Bezos started Amazon with his then-wife MacKenzie Scott, now MacKenzie Bezos.
1995, Creator of the JavaScript programming language is Brendan Eich. He developed the language in just 10 days in May 1995 while working at Netscape Communications Corporation.
2005, Google Maps desktop web application first launched in the U.S.
2008, the app that popularized GPS mapping for the masses was Google Maps, released for Android in 2008 and soon after for iPhone, offering free, real-time, turn-by-turn navigation that changed the smartphone landscape.