Enlightened Mathematics Summer 2026

Seeing numbers through structure

Numbers were represented in multiple geometric and sequential forms rather than only as symbols.

• Number forms: Examples included factor arrays, triangular and tetrahedral numbers, perfect squares, cubes, harmonics, factorials, and powers of two and three.
• Fibonacci sequence: Each term was built visibly from the previous two: 0, 1, 1, 2, 3, 5, 8, 13, and onward.
• Counting in other bases: Binary counting and skip-counting in bases such as 3, 5, 6, and 9 revealed patterns that are hidden when working only in base ten.

Practice with the visual counting tools

Use the course website’s counting tools on a phone or tablet and treat exploration as a regular practice.

• Count Tetris: Skip-count by different amounts, then change the base and observe how the spatial pattern changes.
• Suggested base exercise: Count by twos in base three, and repeat binary counting from zero through 15 or 16.
• Sequence fluency: Practice powers of two through 4096 and factorial counting through 7!.

Tessellations depend on the surrounding space

The polyhedra explorer showed nets, dual constructions, dihedral angles, and solids that fill three-dimensional space without gaps.

• Three-dimensional tessellation: The cube, truncated octahedron, and rhombic dodecahedron were used to illustrate space-filling structures analogous to a two-dimensional honeycomb.
• Flat versus curved space: Pentagons do not tessellate the Euclidean plane, but appropriately sized pentagons can cover spherical space. Hyperbolic space permits patterns, such as seven-sided cells, that do not fit in the flat plane.
• Duals and symmetry: A dual is formed from face-center relationships with vertex positions adjusted so the resulting solid has the required symmetry.

Projecting and constructing a tesseract

A cube’s changing two-dimensional shadow provided the analogy for viewing a fixed four-dimensional object through a changing three-dimensional projection.

• Dimensional construction: Extrusion follows point → line → square → cube → hypercube.
• Tesseract structure: A tesseract is a four-dimensional cube with 16 vertices, 32 edges, and eight cubic cells.
• Coordinates: Its vertices can be described with XYZW coordinates; each coordinate takes a positive or negative value, producing 2⁴ = 16 vertices.
• Rotations: Four-dimensional rotations occur in coordinate planes. There are 6 such rotations in four dimensions, 10 in five dimensions, and 15 in six dimensions—the triangular-number pattern.
• Higher dimensions: Repeated projection loses information but allows higher-dimensional cubes to be displayed through three-dimensional forms and ultimately on a two-dimensional screen.

Possible programming extension

A valuable extension would be to write a program that generates hypercube vertices, connects the correct pairs, and projects the result into lower dimensions. Deriving the projection equations and the counts of vertices, edges, and faces would connect the visual model to computation.

Learning to think mathematically

Understanding was treated as a continuum rather than an all-or-nothing state. The class emphasized staying with the mathematics itself, noticing patterns within its domain, and learning from strong questions—not only collecting answers.

Counting and meditation practices

The opening exercises emphasized direct perception rather than mechanical recitation, and the class ended with a short meditation.

• Direct quantity perception: Count from zero through twelve while directly perceiving each quantity.
• Sequence counting: Practice the Fibonacci sequence and binary counting.
• Closing meditation: After focused mathematical work, sit quietly for about five minutes and notice the lingering quality or “residue” of attention.

Symmetry and duality among polyhedra

The explorer made the defining relationships among the major families visible.

• Platonic solids: Both face-transitive and vertex-transitive: every face is alike and every vertex has the same configuration.
• Archimedean solids: Vertex-transitive but not face-transitive; their vertex configurations describe the same sequence of polygons around every vertex.
• Catalan solids: Duals of the Archimedean solids; they are face-transitive but generally have multiple kinds of vertices.
• Duality: Constructing a dual exchanges faces and vertices while preserving the number of edges, and applying the dual operation again returns to the original family.
• Chirality: Some snub solids have distinct left- and right-handed forms, analogous to left and right hands.
• Nets: A solid can have multiple valid unfoldings rather than one unique net.

Visual arithmetic examples

The revised apps connect symbolic calculations to spatial decompositions.

• Cubing 12: The cube model showed 12³ = 1,000 + 600 + 120 + 8 = 1,728.
• Near-base multiplication: The nearest-base view showed 39² as 40² − 40 − 40 + 1 = 1,521.
• Arithmetic series: Pairing the terms from 1 through 100 gives 100 × 101 ÷ 2 = 5,050.

Questions to investigate further

Several discoveries became starting points for additional study.

• Catalan dual construction: What exact normalization makes the faces congruent when constructing a Catalan solid from an Archimedean dual, and how does this differ from a simple polar dual?
• Three-dimensional tessellation: Which solids in the Platonic, Archimedean, Catalan, and Johnson families can fill Euclidean three-space without gaps, and why?
• Dihedral angles: How do the angles between neighboring faces help explain whether a solid can tessellate space?

App-library development notes

The Polyhedra Explorer supports rotation, zooming, unfolding, dual construction, alternate nets, and separate views of faces, edges, and vertices. Planned improvements include dihedral-angle data, better unit-edge and equal-radius comparison beyond the Platonic solids, and an improved variable-base counting app.

Explore and study the new tools

Open the app library on a computer or phone and experiment with the Polyhedra Explorer and arithmetic visualizations. Payton was specifically asked to look up dihedral angles.

Counting faces, edges, and vertices

Shared vertices and edges must be counted only once. Schläfli symbols encode both the polygonal face and how many faces meet at each vertex.

• Dodecahedron {5,3}: 12 pentagonal faces; V = (12×5)÷3 = 20 and E = (12×5)÷2 = 30.
• Icosahedron {3,5}: 20 triangular faces; V = (20×3)÷5 = 12 and E = (20×3)÷2 = 30.
• Other Platonic solids: Tetrahedron: (F,E,V)=(4,6,4); cube: (6,12,8); octahedron: (8,12,6).

Euler’s characteristic

For any closed mesh topologically equivalent to a sphere, χ = F − E + V = 2. The class verified this for every Platonic solid and discussed why adding an edge that divides a face adds one face as well, leaving χ unchanged.

• Sphere-like topology: No holes: χ=2, even when the mesh is irregular, nonconvex, or has millions of elements.
• Torus: One hole: χ=0. A connected torus decomposition with F=2, E=4, and V=2 gives 2−4+2=0.
• Double torus: Two holes: χ=−2. More generally, each additional hole lowers the characteristic by 2.

Direct perception and self-teaching

The session emphasized seeing mathematical structure rather than carrying formulas as memorized “luggage.”

• Visualization: Count while picturing quantities, binary values, Fibonacci numbers, and the faces, edges, or vertices of solids.
• Simplicity and isolation: When stuck, reduce the problem to the simplest case you can directly inspect.
• Sphere meditation: Picture one vertex and one loop on a sphere. The loop divides the surface into two faces, giving F−E+V = 2−1+1 = 2; then imagine adding vertices and edges while tracking what changes.

The cubic double-torus model

The class built a figure-eight-shaped, seven-segment-style mesh from cubic nodes and connectors. Direct counting produced F=50, E=100, and V=48, so χ=50−100+48=−2, confirming the value for a two-holed surface.

Open question: toroidal analogues of Platonic solids

Is there a finite family of especially regular toroidal polyhedra analogous to the five Platonic solids? The class developed simple cubic torus models, but the broader classification question remained open for further investigation.

Opening number practice

The warm-up emphasized direct perception and mental calculation rather than reciting symbols mechanically.

• Counting: Count from 0 through 12 while visualizing and feeling each quantity.
• Fibonacci sequence: Build each term by adding the previous two, continuing through 987. For example, 55 + 89 can be handled as 55 + 100 − 11 = 144.
• Binary: Practice representing the integers from 0 through 15 in base two.

Dual polyhedra

A dual is constructed by placing a new vertex at the center of every face and connecting centers of adjacent faces. The original solid’s faces become the dual’s vertices, and its vertices become the dual’s faces.

• Cube ↔ octahedron: A cube has 6 faces and 8 vertices; its dual octahedron has 6 vertices and 8 faces.
• Dodecahedron ↔ icosahedron: The 12 pentagonal faces and 20 vertices of the dodecahedron trade with the 12 vertices and 20 triangular faces of the icosahedron.
• Tetrahedron: The tetrahedron is self-dual: constructing its dual produces another tetrahedron.
• Double dual: Taking the dual twice returns the original form, though its scale may change.

Families of three-dimensional forms

The session compared several finite and infinite families of polyhedra.

• Platonic solids: Exactly 5: all faces are congruent regular polygons, and every vertex has the same arrangement.
• Archimedean solids: Exactly 13: regular polygon faces arranged so every vertex has the same configuration. Truncating a cube, for example, produces triangles and octagons with vertex configuration 3-8-8.
• Catalan solids: The 13 duals of the Archimedean solids. Their faces are congruent, though those faces need not be regular polygons.
• Johnson solids: Exactly 92 convex forms built from regular polygon faces after the uniform solids, prisms, and antiprisms are treated separately. A square pyramid is the simplest example discussed.
• Prisms and antiprisms: Prisms connect matching vertices of two parallel polygons with quadrilateral faces. Twisting one polygon produces an antiprism with alternating triangular side faces; these are infinite families.
• Chirality: The snub cube and snub dodecahedron have distinct left- and right-handed versions that become one another in a mirror.

Describing structure with numbers

Vertex configurations and Schläfli symbols compress geometric structure into numerical notation.

• Vertex configuration: List the side counts of the faces meeting at each vertex: 3-8-8 means one triangle and two octagons meet there; 3-4-4-4 means one triangle and three squares.
• Schläfli symbol: For a regular polyhedron, {p,q} means each face is a regular p-gon and q faces meet at every vertex. Duality reverses the two numbers.
• Graphs: A polyhedron can also be understood as a graph of vertices connected by edges. This connects the geometry to information networks and neural-network structures.

Mental construction of forms

Use visualization to understand how the solids are built rather than relying only on names or pictures.

• Truncation: Imagine cutting every corner from a cube at the correct depth. Each removed corner creates a triangle, while each original square becomes an octagon.
• Nets: Visualize a solid unfolded into a two-dimensional arrangement of faces, then mentally fold the net back into the three-dimensional form.
• Form meditation: Imagine being inside a selected form and notice how its proportions, faces, and symmetry affect your spatial experience.

Continue with the Eternal Forms study guide

Keep the provided study-guide PDF and use it to review the Platonic, Archimedean, Catalan, and Johnson solids; prisms and antiprisms; duality; chirality; nets; vertex configurations; and Schläfli symbols.

Number-sense warmups

The opening exercises emphasized familiarity with numerical structures rather than answering a particular problem.

• Visual quantities: Picture numbers as dots in rows, rectangles, or groups—for example, 9 as three groups of three and 10 as two groups of five.
• Sequences and representations: Practice the Fibonacci sequence, binary counting through 15, and repeated doubling: 1, 2, 4, 8, 16, 32, …

Angles of a regular polygon

Moving congruent triangles outward revealed that changing the polygon’s overall size changes its side lengths but not its angles. The exterior angles make one full turn, while each interior angle completes a straight angle with its corresponding exterior angle.

• Exterior angle: For a regular polygon with n sides: E = 360°/n.
• Interior angle: θ = 180° − 360°/n.
• Sum of interior angles: S = n(180° − 360°/n) = 180n − 360 = 180(n − 2).
• Examples: An octagon has exterior angle 45° and interior angle 135°. A pentagon has interior angle 108° and interior-angle sum 540°.
• Radians or tau: Using tau, the exterior angle is τ/n and the interior angle is τ/2 − τ/n.
• Vocabulary: A regular polygon has equal sides and equal angles. Polygons are two-dimensional; polyhedra are three-dimensional forms made from polygons.

Polygon table

Work through regular polygons from 3 through 12 sides. For each one, calculate the exterior angle, one interior angle, and the sum of all interior angles; round repeating decimals to two places. Desmos may be used to organize or verify the calculations, and an optional extension is to rewrite the equations using π or τ.

Week 4: Polygon Triangles, Version 1

Revisit the interactive animation used to separate radial triangles, scale the resulting polygon, and observe how the exterior and interior angles arise without beginning from a memorized formula.

Difference of squares as a mental-math shortcut

Instead of calculating two large squares separately, factor their difference: a² − b² = (a − b)(a + b). This makes specially chosen pairs quick to evaluate.

• 255² − 245²: (255 − 245)(255 + 245) = 10 × 500 = 5,000.
• 105² − 95²: 10 × 200 = 2,000.
• 51² − 49²: 2 × 100 = 200.
• 755² − 745²: 10 × 1,500 = 15,000.

Related films

The teacher highly recommended The Man Who Knew Infinity for its portrayal of Ramanujan, mathematical insight, and the challenge of proof. Cinderella was recommended for its lesson in kindness; V for Vendetta was mentioned for its lesson in integrity, with a warning that it is violent.

Number meditation and fluency

Continue connecting numbers to felt and visual quantities rather than treating them only as symbols.

• Direct perception: Visualize quantities from 0 through 12 as arrangements of dots. Examples included 4 as a tetrahedron, 6 as an octahedron, 8 as a cube’s vertices, 9 as a 3×3 grid, 10 as rows of 1+2+3+4, and 12 as a 3×4 grid or the vertices of an icosahedron.
• Fibonacci sequence: Begin with 0, 1 and repeatedly add the previous two terms, doing the addition mentally when possible.
• Binary counting: A tangible course goal is to count from 0 through 15 in binary, using only 0s and 1s.

Multiplication through areas and offsets

Break each factor into an easy base plus or minus a small amount, then perceive the product as a main rectangle, two side rectangles, and a corner.

• Negative offsets: For 48², use (50−2)²: 2500−100−100+4=2304. The corner is positive because it represents negative 2 times negative 2.
• Overlap interpretation: An equivalent method removes two strips from the larger square and then adds the corner back because it was removed twice.
• Unequal factors: For 56×54, use (50+6)(50+4)=2500+300+200+24=3024. Cross terms must pair each extra part with the other factor’s base.
• Powers: Squaring can reveal higher powers: 49²=7⁴=2401, while 81²=9⁴=3⁸=6561.

Difference of squares

Removing a smaller square from a larger square creates an L-shape that can be rearranged into a rectangle. This makes x²−y²=(x−y)(x+y) visually evident.

• When to use it: Look for two factors equally spaced around a midpoint.
• Examples: 18×22=20²−2²=396; 49×51=50²−1²=2499; and 99×101=100²−1²=9999.
• Geometric meaning: The rearranged rectangle has side lengths equal to the large square’s side plus and minus the small square’s side.

Geometric solids preview

The five Platonic solids were connected back to dot visualization. The cube and octahedron are duals, the dodecahedron and icosahedron are duals, and the tetrahedron is self-dual; the 13 Archimedean solids were mentioned as a future topic.

Schedule change

There will be no Tuesday-night meeting because of travel. Class will resume Wednesday, with the regular Tuesday meeting returning the following week.

Optional practice during the break

Use the free evening for independent meditation, a guided meditation from the course website, multiplication practice with the area models, or rest.

Numerical contemplation and warm-up

The opening practice combined focused attention with direct visualization of quantities and familiar numerical patterns.

• Attention: Generate a sense of light and love while focusing on the heart center, third eye, or navel center.
• Number perception: Count from 0 through 12 while sensing each quantity directly—such as through dots—rather than treating the numeral as only a symbol.
• Sequences and mental images: Review the Fibonacci sequence through 987, binary counting through 15, mental products such as 11³ = 1331 and 21² = 441, and visualizations of a radian and π.
• Ongoing practice: Spend about two minutes a day becoming fluent with the established number sequences so the class warm-up can remain brief and contemplative.

Fibonacci growth and the golden ratio

Fibonacci squares made the recurrence visible: each new side length is the sum of the previous two, and successive-term ratios approach the golden ratio φ.

• Golden-ratio identity: φ² = φ + 1, and more generally φⁿ = φⁿ⁻¹ + φⁿ⁻².
• Limit: The ratio of successive Fibonacci terms approaches φ ≈ 1.618; for example, 987/610 is already very close.
• Base φ exploration: Using powers of φ as place values produces unusual equivalent representations. For example, 110 in base φ equals 1000 in base φ because φ² + φ = φ³.
• Visual spiral: Adjacent squares with side lengths 1, 1, 2, 3, 5, 8, 13, and so on create the familiar Fibonacci spiral construction.

Why the area of a circle is πr²

Two rearrangements showed how curved area can be converted into shapes whose area is easier to calculate.

• Sector rearrangement: Dividing circles into increasingly thin sectors and alternating them approaches a rectangle in the limit. This gives A = ½rC; since C = 2πr, A = πr².
• Concentric-ring rearrangement: Unrolling increasingly thin circular rings forms a triangle with height r and base τr. Its area is ½(τr)(r) = πr².
• π and τ: The diameter fits around the circumference π times, approximately 22/7; the radius fits around it 2π times, also called τ.
• Inscribed circle: A circle inscribed in a square occupies π/4 of the square’s area, or about 78.5%.

Pacing and mathematical language

The teacher agreed to slow down the sequence reviews when needed. Students were also encouraged to say when unfamiliar notation or terminology was blocking the direct mathematical idea, so it could be rebuilt from counting or visualization.

Questions left open for contemplation

The discussion raised broader questions about how mathematical truths depend on—or transcend—the representations used to describe them.

• Alternative foundations: Could area be quantified from a more circular foundation rather than beginning with square units?
• Circle and triangle: Why do both circle-area constructions ultimately form triangles, and how does rotation create triangular structure?
• Dimension: Three non-collinear points define a plane, while four non-coplanar points can define three-dimensional space.

Three ways to understand an angle

The most direct description is a fraction of a turn: a half turn, quarter turn, sixth turn, and so on. Degrees divide a full turn into 360 parts, while one radian is the angle whose arc length equals the radius.

• Degrees: 1° is 1/360 of a full turn. The number 360 is useful because it has many divisors, so common fractional turns have whole-number degree measures.
• Radians: A full turn contains 2π radians, approximately 6.28. One radian is approximately 57.3°.
• Tau: τ = 2π is the number of radii that fit around a circle. Thus τ is one full turn, τ/2 is a half turn, τ/4 is a quarter turn, and τ/6 is a sixth turn.
• Pi: π counts how many diameters fit around a circle. The useful approximation 22/7 can also be understood as 3 + 1/7.

Visual reasoning before memorized formulas

The sum 1 + 2 + ··· + n was reconstructed by joining two staircase-shaped arrays into an n-by-(n+1) rectangle, then taking half: n(n+1)/2. The same emphasis on direct perception was used to decompose squares and cubes—for example, 51² = 2500 + 50 + 50 + 1 = 2601 and 11³ = 1331.

Practices to continue

The goal is to perceive each structure rather than apply an unexplained procedure.

• Number sense: Feel the qualities of 0–12, then visualize each quantity directly as dots or geometric vertices.
• Sequences and bases: Continue the Fibonacci sequence mentally and practice counting in binary until it flows like a familiar verbal sequence.
• Visual sums: Revisit the staircase picture for 1 through 10 and use it to renew why the sum is 55.
• Visual squaring: Practice 61², 71², 81², and 91² by decomposing each square into visible areas.

Repeating decimals and rational approximations

The class observed that 1/7 = 0.142857… and that 142857 × 7 = 999999. Similar repeating blocks arise for other fractions because multiplying the repeating block by its denominator produces a string of nines; this topic will be developed further.

Course reading and discussion

For the foundation of Bhakti mysticism connected with the course, begin with chapters 1–10 of *Futuristic Intelligence*, especially Chapter 1, “A Complete Stereology.” Keep the study active in Circle’s Cohort Chat by asking questions and talking with classmates; the Circle app was recommended for access.

Daily attention and counting practice

Build consistency by practicing at a regular time and place.

• Morning meditation: Meditate for 15 minutes upon waking; heart-centered or light meditation was suggested as a starting point.
• Counting: Regularly count a number sequence such as Fibonacci, binary, or ternary to focus attention and make direct contact with mathematical order.
• Mathematics study: Aim for about 30 minutes of consistent daily study.
• Mindful movement: Walking, running, swimming, or biking can support clarity; walking was especially recommended as a mindfulness practice rather than a time for looking at a phone.

Seeing the binomial cube geometrically

A cube with side length x + a can be separated into an x³ core, square face-layers, linear edge-prisms, and a corner cube. This makes the algebra visible.

• Adding one: (x + 1)³ = x³ + 3x² + 3x + 1. The coefficients 1, 3, 3, 1 correspond to one volume, three faces, three edges, and one corner.
• Adding two: (x + 2)³ = x³ + 6x² + 12x + 8.
• Mental calculation: The picture makes 11³ = 1,331 immediate and also supports calculations such as 12³ = 1,728, 13³ = 2,197, and 21³ = 9,261.
• Connection to bases: The same structure works in other bases: 11₆³ = 1331₆. Each geometric layer supplies a digit associated with a power of the base.

Direct perception rather than mechanical technique

The goal is to recognize foundational mathematical structures so clearly that they become self-evident. Symbols and names help communication, but they are not the quantity or reality they describe; genuine understanding comes from seeing the underlying relationship.

• Mathematical simplicity: Mathematics was presented as internally coherent: if the premises are clear, the consequences follow without opinion.
• Meditation and mathematics: State of mind affects mathematical lucidity. Meditation helps settle scattered attention before turning toward abstraction.

Further viewing and exploration

The class watched Richard Feynman explain the difference between knowing the name of a bird and knowing the bird. The film Arrival was also recommended for its treatment of language, meaning, and communication; the EMSU26 class page was identified as the place to find replays and available mathematical web applications.

• Richard Feynman: Revisit his discussion of knowing names versus knowing things.
• Arrival: Watch for its exploration of how language and differing systems of description shape understanding.

Class communication preference

Send your vote on whether the course chat should be private to this class or open to the wider community.

The central result

Two identical staircases containing 1 + 2 + ⋯ + n units fit together into an n-by-(n + 1) rectangle. Therefore one staircase has half the rectangle’s area: Σᵢ₌₁ⁿ i = n(n + 1)/2.

• Example: 1 through 100: 100 × 101 ÷ 2 = 5,050.
• Example: 1 through 10: 10 × 11 ÷ 2 = 55.

Algebra expresses a perceived structure

Arithmetic works with specific numbers; algebra generalizes a pattern using variables. An equation places two different expressions of the same value side by side—for example, a long sum on one side and the dimensions of a rectangle divided by two on the other.

• Summation notation: Sigma notation compactly instructs us to substitute successive values and add the resulting terms.
• Equivalent forms: n(n + 1) and n² + n describe the same rectangle; similarly, two staircases equal one n-by-(n + 1) rectangle.

Extensions: squares and arithmetic sequences

The sum of consecutive squares was pictured as a layered pyramid. Six copies can be rearranged into a rectangular prism, giving Σᵢ₌₁ⁿ i² = n(n + 1)(2n + 1)/6; for n = 10, the sum is 385. For an arithmetic sequence, aₙ = a₁ + (n − 1)d and Sₙ = n(a₁ + aₙ)/2; for example, 3 + 5 + 7 + 9 + 11 = 35.

Contemplative practice

The opening practice located the heart center near the middle of the sternum, invoked warmth and energy there, and shifted attention so consciousness seemed to look out from the heart. Students were also asked to try the 17-minute “Four Modes of Counting” guided contemplation and were encouraged to begin with the cessation meditation unless another practice felt more suitable.

• Learning attitude: Remain with uncertainty rather than rushing to escape it; allow the structure to become clear through sustained attention.

Big-O comparison for Payton

Determine the Big-O complexity of summing 1 through n with a loop and compare it with evaluating n(n + 1)/2 directly.

Coding and class communication

For extracurricular study, implement both the brute-force loop and the direct formula in Python, JavaScript, or Java and compare their results and efficiency. Students were encouraged to use the Circle app with notifications enabled, post course questions in the Hall of Inquiry, and begin interacting in the class space; a dedicated class chat was to be created.

Multiplication as direct perception

Rectangular area models made products visible by separating tens and ones. For example, 21² becomes 400 + 20 + 20 + 1 = 441, while 21 × 22 becomes 462. The same quantity can also be reorganized: 6 × 6 can be seen as four groups of nine or nine groups of four, and 12 × 12 as 4 × 36 or 9 × 16.

• Examples practiced: 21² = 441; 21 × 22 = 462; 31² = 961; 71² = 5,041; 11 × 12 = 132; 13 × 12 = 156; 14² = 196.

From triangular sums to arithmetic series

Two copies of 1 + 2 + ··· + n form an n-by-(n+1) rectangle, giving 1 + 2 + ··· + n = n(n+1)/2. More generally, an arithmetic series can be found by pairing its first and last terms: S = (a₁ + aₙ)N/2, where N is the number of terms and aₙ = a₁ + (N−1)d.

• Consecutive integers: 1 + 2 + ··· + 100 = 100 × 101 ÷ 2 = 5,050.
• Different starting point: 3 + 4 + 5 + 6 + 7 = (3 + 7) × 5 ÷ 2 = 25.
• Common difference of three: 3 + 6 + 9 + ··· + 120 has 40 terms, so its sum is (3 + 120) × 40 ÷ 2 = 2,460.
• Summation notation: The lower and upper values specify which term numbers to use; the expression beside Σ says what to evaluate for each of those values before adding the results.

Mathematics beyond utility

Mathematics was presented as playful inquiry in abstraction—something that can refine attention, clarity, and perception much as poetry, music, or meditation can. The emphasis was on slowing down enough to understand relationships directly rather than memorizing formulas or rushing through material.

Number and attention practices

Continue the four-part counting warm-up and use meditation to notice how mathematical structures remain present in the mind afterward.

• Feel the numbers: Count slowly and experience the character of each number rather than attending only to its symbol.
• Visualize quantities: Represent numbers directly with dots or spatial arrangements—for example, three as a triangle and eight as the corners of a cube.
• Sequence counting: Build Fibonacci numbers by adding the previous two terms. Binary counting or counting by a chosen interval in another base can provide an additional challenge.
• Meditate after mathematics: Sit quietly and notice the mathematical structures “echoing” in the mind and any increased strength or steadiness of attention.

Weekend study

Rewatch and actively study the lesson rather than trying merely to finish it.

• Visual multiplication: Practice teen-by-teen products with area models until the parts can be perceived mentally.
• Arithmetic series: Reconstruct the formulas from the rectangle or paired-term picture, then practice examples such as 3 + 6 + 9 + ··· + 120.
• Personalized counting: Kimberly was encouraged to practice Fibonacci counting; Scott was encouraged to continue binary counting or take it to a more challenging level.

Desmos summation notation

Typing “sum” in Desmos produces the summation template. It was used to enter lower and upper bounds, evaluate expressions such as n or n² across those bounds, and check arithmetic-series calculations.

Four counting warm-ups

Use these to engage the mind while developing intuition for number.

• Feel number: Count from 0 to 12 while attending to the distinct quality or “ness” of each number, without needing to visualize it.
• Perceive number directly: Count again while seeing quantities rather than numeral symbols—for example, three dots or a triangle for 3, and symmetric forms for larger numbers.
• Follow a sequence: Generate the Fibonacci sequence by repeatedly adding the final two terms: 0, 1, 1, 2, 3, 5, 8, 13, 21…
• Change the base: Practice counting in base 3: 0, 1, 2, 10, 11, 12, 20, 21, 22, 100…

Base conversion shortcuts

A numeral such as 1000 in base b represents b³. One less than a power of the base is written using repeated copies of that base’s highest digit.

• Base 16: 255 in base 10 is FF in base 16 because 256 is 100₁₆ and 255 is one less.
• Binary: 7 is 111₂; similarly, 31 is five 1s because 32 is 1 followed by five zeros in binary.
• Base 8: 16 in base 10 is 20₈, meaning two groups of eight and no remainder.

Seeing multiplication and exponents

Square products were decomposed visually around a convenient base. For example, 51² becomes 50² + 50 + 50 + 1 = 2,601, while 1,024² becomes 1,000,000 + 24,000 + 24,000 + 576 = 1,048,576.

• Adding exponents: When powers with the same base are multiplied, their exponents add: 2⁵ × 2⁵ = 2¹⁰.
• Representation changes difficulty: 2¹⁰ is 1,024 in base 10, but in binary it is simply 1 followed by ten zeros. In base 4, the same quantity is 1 followed by five zeros.
• Learning strategy: When a question feels inaccessible, return to the smallest known cases, notice the pattern, and proceed gently rather than forcing an answer.

Prime numbers and the Ulam spiral

Prime numbers have no positive divisors other than 1 and themselves. Plotting primes while spiraling the integers produces the Ulam spiral, where unexpected diagonal patterns appear; a scene from the film Contact illustrated why primes can serve as a recognizable sign of intelligence.

Meditation and study rhythm

The suggested daily rhythm was 15 minutes of light meditation followed by 15–45 minutes of mathematical practice, ideally totaling about an hour when possible. Guided meditations may be used, and the emphasis is on practicing only while reconnecting with curiosity and play.

• Group meditation: A light meditation was planned for July 14, 2026, from 6:00–7:00 PM Pacific. Reviewing the first sermon meditation beforehand was suggested if time allowed.

Base-conversion exercises

Optional exercises available in the practice contemplations were assigned for further practice.

• Exercise 1: Convert 111 from base 3 to base 10.
• Exercise 2: Convert 101 from base 9 to base 10.

Recommended viewing and reading

The film Contact was highly recommended. Named books included Art of Loving by Erich Fromm, On Love and Loneliness by J. Krishnamurti, As a Man Thinketh, Education and the Significance of Life, and a Bhagavad Gita translation by Paramahansa Yogananda.

Fractal preview

An interactive Julia-set graphic briefly demonstrated how repeating simple operations can produce intricate fractal patterns. A fuller treatment was left for a later class.

Place value in different bases

Each position represents a power of the chosen base. When all available digits have been used, counting carries into the next position—just as base ten moves from 9 to 10.

• Binary: The positions represent 1, 2, 4, 8, 16, and so on. A 1 turns that power of two on; a 0 turns it off. For example, 110₂ is 4 + 2 = 6.
• Base eight: Digits run from 0 through 7. Thus 77₈ is 63 in base ten, and the next number is 100₈, or 64.
• Gentler starting point: Base eight was recommended as an easier way to become comfortable with unfamiliar place-value systems before returning to binary.

Fractional positions use reciprocal powers

Places to the right of the point divide repeatedly by the base. This makes 0.1₂ equal to 1/2, 0.01₂ equal to 1/4, and 0.001₂ equal to 1/8; similarly, 0.4₈ equals 1/2.

• Representations depend on the base: A fraction that repeats in one base may terminate in another. One-third repeats in base ten but is a single digit, 0.4, in base twelve.
• Repeating eximals: Examples included 1/9 = 0.111… in base ten, 1/3 = 0.010101… in binary, and 1/2 = 0.111… in base three.
• Visual meaning: The interactive diagrams subdivided an area by the base at each level. The colored regions showed directly how the infinite expansion fills exactly the stated fraction.

Optional mathematical practice

Full mastery was not required before the next class, which will be largely independent of this material. Students were encouraged to deepen whichever example felt most compelling.

• Contemplate one pattern: Try holding a single visualization in mind, such as one-third in binary or one-half in base three, and notice how each successive subdivision completes the fraction.
• Look for structure: Count in another base, write the corresponding place values, and identify which powers are active rather than treating the written digits as base-ten numbers.
• Study together: Students may message one another through Circle to practice, teach the idea to a partner, or explore more challenging patterns.

Daily light practice

A 15-minute daily practice was recommended, preferably in the morning before speaking with anyone. The suggested method was gentle gazing with a mirror, crystal, or candle while sitting upright; even one clear minute of experiencing light within the session was described as valuable.

Interactive eximals app

The class app is available in Circle under “Sanctum of Mathematical Truth” → “Week One” through the Google Drive software folder. Download all files into one folder, open the main file beginning with “app_” in Processing, and press Play; the spacebar switches between automatic animation and manual mode.

Must this be mastered before the next class?

No. The lesson was presented as an introduction that planted the core ideas. Further practice is optional and can continue throughout the course.

Numerals depend on their base

A written numeral does not have a complete meaning until its base is known. Each position represents a power of that base, and appending a zero multiplies the value by the base.

• Base 5: After 4 comes 10₅, meaning one group of five and no remainder. Thus 44₅ is followed by 100₅, and 100₅ equals 25 in base 10.
• Base 16: 17 in base 10 is written 11₁₆: one group of sixteen and one more. Hexadecimal commonly uses A–F for values 10–15.
• Binary: Binary uses only 0 and 1. Adding a zero multiplies by 2, and 65 in base 10 is 1000001₂.
• Digit length: Larger bases generally need fewer digits to represent the same value; binary often requires many more digits.

Develop multi-numeral fluency

Practice seeing quantities directly in their new base rather than repeatedly converting through base 10. The teacher suggested making mathematics a regular contemplative practice, even for 15 minutes each day.

• Count aloud: Count in base 5 and binary until transitions such as 44₅ → 100₅ and 111₂ → 1000₂ feel natural.
• Use explicit language: Read 14₅ as “one five and four,” rather than “fourteen,” to keep the grouping visible.
• Explore doubling: Use powers of two to construct binary values: 1, 2, 4, 8, 16, 32, 64, and then combine them.

Meditation and continued study

Students were invited to use the class community throughout the week for discussion and questions, and to explore its short cessation and light meditations and its collection of sermons. The next meditation session was announced for July 7, 2026, from 6:00–7:00 p.m. Pacific, with attendance encouraged when possible.