Enlightened Mathematics Summer 2025

In our journey into mathematics, we will begin by elucidating the very foundations of quantification: counting. We will bring clarity to the actual nature of the base-10 counting system ubiquitous to humanity and expand upon it by exploring other base systems, such as base-12, binary system, and hexadecimal.

Looking at novel base systems acts as a pedagogical exercise in developing the ancient Zen principle of beginner’s mind. By learning to perceive math in novel bases, we don’t rely on memory, but rather develop the ability to see mathematics anew, directly, and with fresh eyes. Even in the realm of the most basic form of mathematics, counting, there is so much to learn!

We expand counting and arithmetic operations in novel bases to something beautifully exotic: fractional representations in novel floating-point systems, that is, and how to express fractions in bases other than base 10.

Exploring novel base systems distills mathematics of human artifact, allowing us to begin to see it in its cosmic essence. To see the truth in its actuality is an exercise of the true philosopher and student of enlightenment. To seek to experience quantity without the imposition of humanity is to enter into the eternal realm, to commune with the stars, to experience truth. By changing bases, we emancipate ourselves from the limitations of perception and transcend perspective.

We expand upon our inquiry into bases by introducing the mathematics of sequences and series—lists of numbers that evolve according to particular arithmetic rules.

We will contemplate these abstractions visually through the geometric lens, making the equations that describe sequences and series directly perceivable and self-evident.

This inquiry will allow us to expand upon our inquiry into novel bases by helping us see the nature of repeating decimals in novel bases and in base-10 itself.

This week, we inquire into the generalization of arithmetic and the introduction of variables.

We learn to see fundamental algebraic equations as self-evident truths through visual, geometric representation, starting with perception and moving to description that is symbolic.

The concepts that we will study include differences of squares, algebraic factoring, and even polynomial long division.

Upon which we turn back around and look to the arithmetic foundations from whence we came, and see that our deep comprehension of algebra can be applied to awaken elegant capacity in mental mathematics.

Life itself is about seeing that which exists beyond the intellect, and then describing it, allowing the epistemological perception to remain distilled of conception. This is the knowing of union, the knowing of love.

Inquire into the cosmic forms within the field of dimensionality. Platonic archetypes existing unmanifest, and yet as templates for the manifest.

We will explore fundamental geometric concepts with an enlightened degree of lucidity, considering the nature of angles and how to quantify them.

We reveal the quantities of pi and tau, and foundational equations of geometry, including the circumference and area of a circle.

We will consider the polygon and describe it through the language of mathematics—through equations that are subjectively aligned with our own comprehension.

We will move into the realm of three-dimensional geometry, communing with a direct knowledge of the nature of spheres, pyramids, and cones, and letting that knowledge be semantically described with the symbolism of mathematics.

Existing like beautiful wildflowers in the fields of dimensionality are five 3-dimensional forms known as the Platonic solids, and thirteen 3-dimensional forms known as the Archimedean solids.

We will explore the profundity of these forms and their unique, finite expressions.

We will take special note of one in particular: the icosahedron, and the presence of the golden ratio imbued within its form.

Then, we transcend into the 4th dimension. We will explore some of the wildflowers within that realm, including the tesseract (the 4-dimensional cube), polytopes (4-dimensional analogs of the Platonic solids), stereographic projection, which are like 3-dimensional shadows of 4-dimensions forms, and even the exotic 4-dimensional flat torus.

We will touch on manifolds and their presence in Einsteinian spacetime, and consider the Poincaré Conjecture, one of the most profound and mysterious questions in modern mathematics.

This week expands our spatial consciousness beyond the manifest and its dimensional limitations. What you focus on, you become—and when we focus on the perfection of these existential shapes, the mind resonates as the perfection of these existential archetypes.

Entering into a new plane of quantification, we embark upon the rigors known conventionally by 21st-century humanity as trigonometry and complex numbers.

We begin with the direct perception of sine and cosine, and expand to their related manifestations in the full set of six trigonometric functions. We will deepen our understanding of these mathematical abstractions by exploring their expression as infinite series—a form of mathematics where algebraic polynomials become infinite in expression.

To prepare for our launch into the stars of Euler’s formula, we will explore perhaps one of the most exotic gifts in all of mathematics: imaginary numbers and the complex plane. Through the integration of imaginary numbers and trigonometry, we will perceive and express what is often considered the most beautiful equation in all of mathematics—Euler’s formula.

We will also introduce a new universal number: the number e, perceiving its simple nature, and taking note of its presence in complex rotation and exponential growth.

With our recent entrée into the realm of complex numbers, our next natural step includes a certain family of fractals—the Mandelbrot and the Julia set.

This week, we implement programmatic computation as an indispensable mathematical tool for the futuristic mathematician. Through very fundamental and basic coding, we will manifest what heretofore (recent 50 years) has been veiled to the eyes of man: the computationally derived fractal.

Exploring yet another existential form in the complex plane, we introduce the concept of factorials and then their generalization in complex numbers—this is our inquiry into the Gamma function.

This week bridges the ancient mathematicians and their mind-bound wonder with the unprecedented technological tools of today.

Revealing the self-evident nature of Calculus—and without years of foundational development.

We will emphasize seeing calculus directly, beyond the symbol. The Fundamental Theorem of Calculus and key equations—both in derivatives and integrals—will become self-evident, not as rules to memorize, but as perceptual truths.

You’ll perceive the power rule intuitively. You’ll see—as self-evident—the product rule and the chain rule, arising naturally from your direct perception.

With your new comprehension of the universal number e, you’ll even be able to directly perceive the integral of 1/x​, and its solution involving the natural logarithm.

Calculus opens a new world—one of infinite expression. It grants our mathematical mind a deeper freedom—to move through the infinite, to flow with form.

Enter the matrix—a new dimension of mathematics that empowers many of the most advanced fields in modern technology.

Through direct perception and geometric emphasis, we will learn the nature of vectors and matrices, including eigenvectors and eigenvalues, not as abstract definitions, but as self-evident truths that can be seen, felt, and understood from within.

This week will open the world of linear algebra not through rote mechanics, but as a living system of geometric transformation and dimensional insight.

We will also explore how linear algebra forms the foundation of artificial intelligence, including gradient descent and the ways machines learn through transformation in multidimensional space.