One application, one module, one shape, one journey — rotated daily.https://lemma.wiki/en-UShttps://lemma.wiki/en/finance/bitcoin-signaturelemma:20587:application:bitcoin-signatureAnyone can see your Bitcoin address. No one should see your private key. So how can the network verify that you authorized a payment without learning the secret that authorizes all payments?Thu, 14 May 2026 00:00:00 GMTapplicationfinancehttps://lemma.wiki/en/modules/distributionslemma:20587:module:distributionsA probability is one guess. A distribution is the whole shape of uncertainty. The thing softmax produces, the thing a histogram is, the thing a return is drawn from. Most quantities a model predicts or a portfolio holds are distributions before they are numbers.Thu, 14 May 2026 00:00:00 GMTmodulemodulehttps://lemma.wiki/en/shapes/equilibriumlemma:20587:shape:equilibriumTwo opposing forces, one point where they cancel. The shape underneath the pendulum, the falling raindrop, and the optimal portfolio: name the forces, write the balance equation, solve for where the net change vanishes.Thu, 14 May 2026 00:00:00 GMTshapeshapehttps://lemma.wiki/en/journey/where-change-vanisheslemma:20587:journey:where-change-vanishesOne question — *where does change vanish?* — under three physics applications.Thu, 14 May 2026 00:00:00 GMTjourneyjourney · 4 dayshttps://lemma.wiki/en/finance/portfolio-risklemma:20586:application:portfolio-riskTwo risky assets, mixed, can be less risky than either alone. The mathematics is one cross-term in a quadratic — and it's the same identity that makes a vector sum's length more (or less) than the sum of its parts. Variance, covariance, correlation; the dot product of returns.Wed, 13 May 2026 00:00:00 GMTapplicationfinancehttps://lemma.wiki/en/modules/entropylemma:20586:module:entropy20 questions to find any one of N items needs log₂ N when items are equally likely. Entropy is the generalization — the expected number of yes/no questions when they aren't. The bound everything from Wordle to Huffman to password strength bumps against.Wed, 13 May 2026 00:00:00 GMTmodulemodulehttps://lemma.wiki/en/shapes/find-the-minimumlemma:20586:shape:find-the-minimumThe nouns change — loss, calibration error, portfolio risk — but the procedure is one: pick a number to improve, move through choices, stop when improvement runs out. Once you read it once, you recognise it everywhere.Wed, 13 May 2026 00:00:00 GMTshapeshapehttps://lemma.wiki/en/journey/how-compression-workslemma:20586:journey:how-compression-worksOne three-step procedure under JPEG, TF-IDF, and the raw-pixel entropy floor — across graphics and ML.Wed, 13 May 2026 00:00:00 GMTjourneyjourney · 4 dayshttps://lemma.wiki/en/graphics/bezier-curveslemma:20585:application:bezier-curvesA designer drags four handles. A movie character gets a smooth cheek, a car gets a perfect hood, a letter gets its curve. How does a computer turn a few points into a smooth path?Tue, 12 May 2026 00:00:00 GMTapplicationgraphicshttps://lemma.wiki/en/modules/vectorslemma:20585:module:vectorsA point says where. A vector says how to move. The same tuple plays four roles — position, displacement, velocity, feature — across graphics, physics, and ML. Two operations (add and scale) carry every one of them.Tue, 12 May 2026 00:00:00 GMTmodulemodulehttps://lemma.wiki/en/shapes/compressionlemma:20585:shape:compressionHigh-dimensional data with too much redundancy. Change basis until most coordinates are small. Drop the small ones. Reconstruct what's left. JPEG, TF-IDF, and Huffman codes are the *same procedure*, just applied to pixels, words, and probabilities.Tue, 12 May 2026 00:00:00 GMTshapeshapehttps://lemma.wiki/en/journey/vectors-everywherelemma:20585:journey:vectors-everywhereSame tuple, four roles — across graphics, physics, ML, and finance.Tue, 12 May 2026 00:00:00 GMTjourneyjourney · 5 dayshttps://lemma.wiki/en/graphics/image-compressionlemma:20584:application:image-compressionTwo same-size images: one shrinks, one doesn't. The histogram tells half the story; the spatial structure tells the rest. The entropy module names the floor — this page shows what that floor isn't bounding.Mon, 11 May 2026 00:00:00 GMTapplicationgraphicshttps://lemma.wiki/en/modules/linearizationlemma:20584:module:linearizationMost equations are hard. Their tangent line at a point is easy. Replace one with the other and you get a tool that powers the pendulum clock, Newton's method, and gradient descent — valid in a regime, wrong outside it. The discipline is the regime.Mon, 11 May 2026 00:00:00 GMTmodulemodulehttps://lemma.wiki/en/shapes/inverse-problemlemma:20584:shape:inverse-problemThe forward model is easy: given inputs, compute the output. The inverse is the question that actually matters: given the output, recover the input. Most of finance and most of model calibration is the inverse direction of an equation that's trivial forward.Mon, 11 May 2026 00:00:00 GMTshapeshapehttps://lemma.wiki/en/journey/finding-the-minimumlemma:20584:journey:finding-the-minimumOne skeleton — objective, move, step, stop — under three applications across ML and finance.Mon, 11 May 2026 00:00:00 GMTjourneyjourney · 4 dayshttps://lemma.wiki/en/graphics/jpeg-compressionlemma:20583:application:jpeg-compressionLossy compression isn't bound by entropy — it picks what to discard. JPEG changes basis (DCT) so the picture becomes sparse, throws away the coordinates that don't matter (quantization), and Huffman-packs the rest. Three steps, one savings: fewer coefficients, smaller values, longer zero runs.Sun, 10 May 2026 00:00:00 GMTapplicationgraphicshttps://lemma.wiki/en/modules/integrationlemma:20583:module:integrationA speedometer reads. How far have you traveled? The integral is what survives when you sum a rate over time. The pair to the derivative — and the fundamental theorem says they are inverses.Sun, 10 May 2026 00:00:00 GMTmodulemodulehttps://lemma.wiki/en/shapes/equilibriumlemma:20583:shape:equilibriumTwo opposing forces, one point where they cancel. The shape underneath the pendulum, the falling raindrop, and the optimal portfolio: name the forces, write the balance equation, solve for where the net change vanishes.Sun, 10 May 2026 00:00:00 GMTshapeshapehttps://lemma.wiki/en/journey/change-and-accumulationlemma:20583:journey:change-and-accumulationHow calculus runs both physics and finance — same math, different nouns.Sun, 10 May 2026 00:00:00 GMTjourneyjourney · 7 dayshttps://lemma.wiki/en/graphics/curve-intersectionslemma:20582:application:curve-intersectionsDrag two conics. Sometimes you see four crossings. Sometimes two. Sometimes none. Bezout says the count is still four. Where did the missing intersections go — and what does a graphics engine do about it?Sat, 09 May 2026 00:00:00 GMTapplicationgraphicshttps://lemma.wiki/en/modules/bezoutlemma:20582:module:bezoutTwo curves of degrees d, e meet in exactly d·e points — once the plane is repaired three ways. The chord-and-tangent feeds elliptic-curve arithmetic, which feeds Bitcoin signatures.Sat, 09 May 2026 00:00:00 GMTmodulemodulehttps://lemma.wiki/en/shapes/find-the-minimumlemma:20582:shape:find-the-minimumThe nouns change — loss, calibration error, portfolio risk — but the procedure is one: pick a number to improve, move through choices, stop when improvement runs out. Once you read it once, you recognise it everywhere.Sat, 09 May 2026 00:00:00 GMTshapeshapehttps://lemma.wiki/en/journey/to-backproplemma:20582:journey:to-backpropWhy does a model train? It walks downhill. The hill is a function.Sat, 09 May 2026 00:00:00 GMTjourneyjourney · 7 dayshttps://lemma.wiki/en/physics/projectile-motionlemma:20581:application:projectile-motionThrow a ball. Ignore air. Its horizontal motion keeps time; its vertical motion loses to gravity. Why does that make a parabola?Fri, 08 May 2026 00:00:00 GMTapplicationphysicshttps://lemma.wiki/en/modules/derivativeslemma:20581:module:derivativesA moving point leaves a trail. The derivative is not the trail — it is the arrow the trail wants to become at this instant. Secant slopes converge to tangent slopes, and the same machine becomes slope, velocity, and rate.Fri, 08 May 2026 00:00:00 GMTmodulemodulehttps://lemma.wiki/en/shapes/compressionlemma:20581:shape:compressionHigh-dimensional data with too much redundancy. Change basis until most coordinates are small. Drop the small ones. Reconstruct what's left. JPEG, TF-IDF, and Huffman codes are the *same procedure*, just applied to pixels, words, and probabilities.Fri, 08 May 2026 00:00:00 GMTshapeshapehttps://lemma.wiki/en/journey/to-bitcoinlemma:20581:journey:to-bitcoinFrom the most expensive pizza in history to the signature that keeps a wallet safe.Fri, 08 May 2026 00:00:00 GMTjourneyjourney · 7 dayshttps://lemma.wiki/en/physics/terminal-velocitylemma:20580:application:terminal-velocityGravity pulls forever, but a falling raindrop doesn't speed up forever. The reason is one first-order equation — a derivative balanced against a force that grows with speed. Terminal velocity is not a maximum; it's an equilibrium.Thu, 07 May 2026 00:00:00 GMTapplicationphysicshttps://lemma.wiki/en/modules/parametric-curveslemma:20580:module:parametric-curvesA curve is not a picture. Three motions can paint the same parabola — same image, different parametrizations. Pin down the distinction the word 'curve' was hiding, and a whole stack of downstream tools snaps into place.Thu, 07 May 2026 00:00:00 GMTmodulemodulehttps://lemma.wiki/en/shapes/inverse-problemlemma:20580:shape:inverse-problemThe forward model is easy: given inputs, compute the output. The inverse is the question that actually matters: given the output, recover the input. Most of finance and most of model calibration is the inverse direction of an equation that's trivial forward.Thu, 07 May 2026 00:00:00 GMTshapeshapehttps://lemma.wiki/en/journey/working-backwardlemma:20580:journey:working-backwardThree pages, one question — given the output, find the input.Thu, 07 May 2026 00:00:00 GMTjourneyjourney · 4 dayshttps://lemma.wiki/en/physics/pendulum-clocklemma:20579:application:pendulum-clockDouble the swing. Why does the clock barely change? A real pendulum's period depends on amplitude — but for small swings it almost doesn't. Linearize sin θ ≈ θ, and the period becomes constant. The clock stands on that approximation.Wed, 06 May 2026 00:00:00 GMTapplicationphysicshttps://lemma.wiki/en/modules/loglemma:20579:module:logThe trick that turns × into +. The whole module is one equation; everything else is consequence.Wed, 06 May 2026 00:00:00 GMTmodulemodulehttps://lemma.wiki/en/shapes/equilibriumlemma:20579:shape:equilibriumTwo opposing forces, one point where they cancel. The shape underneath the pendulum, the falling raindrop, and the optimal portfolio: name the forces, write the balance equation, solve for where the net change vanishes.Wed, 06 May 2026 00:00:00 GMTshapeshapehttps://lemma.wiki/en/journey/where-change-vanisheslemma:20579:journey:where-change-vanishesOne question — *where does change vanish?* — under three physics applications.Wed, 06 May 2026 00:00:00 GMTjourneyjourney · 4 dayshttps://lemma.wiki/en/physics/damped-oscillatorlemma:20578:application:damped-oscillatorA pendulum left alone slows and stops; the same pendulum pushed on the beat swings higher and higher. One equation — ẍ + 2γ·ẋ + ω₀²·x = F(t) — runs both. Damping decides how it stops; resonance decides when an outside push wins.Tue, 05 May 2026 00:00:00 GMTapplicationphysicshttps://lemma.wiki/en/modules/optimizationlemma:20578:module:optimizationOptimization is not finding the formula. It is choosing a quantity to improve, then moving through possible choices until improvement stops. The same five steps — objective, search space, move, step size, stopping — run gradient descent in ML, weight selection in portfolios, and temperature fitting in calibration. Different algorithms, identical bones.Tue, 05 May 2026 00:00:00 GMTmodulemodulehttps://lemma.wiki/en/shapes/find-the-minimumlemma:20578:shape:find-the-minimumThe nouns change — loss, calibration error, portfolio risk — but the procedure is one: pick a number to improve, move through choices, stop when improvement runs out. Once you read it once, you recognise it everywhere.Tue, 05 May 2026 00:00:00 GMTshapeshapehttps://lemma.wiki/en/journey/how-compression-workslemma:20578:journey:how-compression-worksOne three-step procedure under JPEG, TF-IDF, and the raw-pixel entropy floor — across graphics and ML.Tue, 05 May 2026 00:00:00 GMTjourneyjourney · 4 dayshttps://lemma.wiki/en/finance/bitcoin-pizzalemma:20577:application:bitcoin-pizzaOn May 22, 2010, a Florida programmer paid 10,000 BTC for two Papa John's pizzas — about $41. Sixteen years later, those coins are worth $1 billion. The most expensive meal in history.Mon, 04 May 2026 00:00:00 GMTapplicationfinancehttps://lemma.wiki/en/modules/distributionslemma:20577:module:distributionsA probability is one guess. A distribution is the whole shape of uncertainty. The thing softmax produces, the thing a histogram is, the thing a return is drawn from. Most quantities a model predicts or a portfolio holds are distributions before they are numbers.Mon, 04 May 2026 00:00:00 GMTmodulemodulehttps://lemma.wiki/en/shapes/compressionlemma:20577:shape:compressionHigh-dimensional data with too much redundancy. Change basis until most coordinates are small. Drop the small ones. Reconstruct what's left. JPEG, TF-IDF, and Huffman codes are the *same procedure*, just applied to pixels, words, and probabilities.Mon, 04 May 2026 00:00:00 GMTshapeshapehttps://lemma.wiki/en/journey/vectors-everywherelemma:20577:journey:vectors-everywhereSame tuple, four roles — across graphics, physics, ML, and finance.Mon, 04 May 2026 00:00:00 GMTjourneyjourney · 5 dayshttps://lemma.wiki/en/finance/present-valuelemma:20576:application:present-valueA future dollar is not a present dollar. Discounting and integration price every cash-flow stream — from rent to bonds to perpetuities — through one identity: future cash times the discount factor, summed across time.Sun, 03 May 2026 00:00:00 GMTapplicationfinancehttps://lemma.wiki/en/modules/entropylemma:20576:module:entropy20 questions to find any one of N items needs log₂ N when items are equally likely. Entropy is the generalization — the expected number of yes/no questions when they aren't. The bound everything from Wordle to Huffman to password strength bumps against.Sun, 03 May 2026 00:00:00 GMTmodulemodulehttps://lemma.wiki/en/shapes/inverse-problemlemma:20576:shape:inverse-problemThe forward model is easy: given inputs, compute the output. The inverse is the question that actually matters: given the output, recover the input. Most of finance and most of model calibration is the inverse direction of an equation that's trivial forward.Sun, 03 May 2026 00:00:00 GMTshapeshapehttps://lemma.wiki/en/journey/finding-the-minimumlemma:20576:journey:finding-the-minimumOne skeleton — objective, move, step, stop — under three applications across ML and finance.Sun, 03 May 2026 00:00:00 GMTjourneyjourney · 4 dayshttps://lemma.wiki/en/ml/tf-idflemma:20575:application:tf-idfWhy does Google show those results in that order? For thirty years the decision rule barely changed. TF-IDF ranks documents by a score that looks like a sum of probabilities — and the log inside it is the same log as in entropy. Rare words carry many bits; common words carry near zero. Stopwords aren't a list, they're the zero set of a function.Sat, 02 May 2026 00:00:00 GMTapplicationml / dlhttps://lemma.wiki/en/modules/vectorslemma:20575:module:vectorsA point says where. A vector says how to move. The same tuple plays four roles — position, displacement, velocity, feature — across graphics, physics, and ML. Two operations (add and scale) carry every one of them.Sat, 02 May 2026 00:00:00 GMTmodulemodulehttps://lemma.wiki/en/shapes/equilibriumlemma:20575:shape:equilibriumTwo opposing forces, one point where they cancel. The shape underneath the pendulum, the falling raindrop, and the optimal portfolio: name the forces, write the balance equation, solve for where the net change vanishes.Sat, 02 May 2026 00:00:00 GMTshapeshapehttps://lemma.wiki/en/journey/change-and-accumulationlemma:20575:journey:change-and-accumulationHow calculus runs both physics and finance — same math, different nouns.Sat, 02 May 2026 00:00:00 GMTjourneyjourney · 7 dayshttps://lemma.wiki/en/ml/model-calibrationlemma:20574:application:model-calibrationA model says '70% confident.' Does that mean, across many such predictions, seven in ten are right? The number on the screen and the long-run frequency are two different quantities. The reliability diagram makes the gap visible; one scalar — temperature — rotates the curve back to the diagonal.Fri, 01 May 2026 00:00:00 GMTapplicationml / dlhttps://lemma.wiki/en/modules/linearizationlemma:20574:module:linearizationMost equations are hard. Their tangent line at a point is easy. Replace one with the other and you get a tool that powers the pendulum clock, Newton's method, and gradient descent — valid in a regime, wrong outside it. The discipline is the regime.Fri, 01 May 2026 00:00:00 GMTmodulemodulehttps://lemma.wiki/en/shapes/find-the-minimumlemma:20574:shape:find-the-minimumThe nouns change — loss, calibration error, portfolio risk — but the procedure is one: pick a number to improve, move through choices, stop when improvement runs out. Once you read it once, you recognise it everywhere.Fri, 01 May 2026 00:00:00 GMTshapeshapehttps://lemma.wiki/en/journey/to-backproplemma:20574:journey:to-backpropWhy does a model train? It walks downhill. The hill is a function.Fri, 01 May 2026 00:00:00 GMTjourneyjourney · 7 days