TESSERA Open the app →

Build your first equations

Tessera is a visual manipulator for tensor algebra: expressions are built by fitting blocks together, the way a mosaic is laid. This page walks through four equations, from two blocks clicked into place to one the program simplifies for you, and ends with four more worth building. The first three take about five minutes.

The metaphor is mathematical, not geometric: what you see is the formal structure of a tensor expression — not a manifold, a curve or a field. The geometry of a block is the signature of the tensor, and checking algebraic validity is collision detection: if an expression is not well formed, the blocks do not fit.

The model is T-mosaic, from Giancarlo Bernacchi's Tensors Made Easy. Section marks (§) here and in the program point into that book; they are never translated, because they say where to read, not what to read. Open Tessera in another tab and follow along.

FIRST The block

Two blocks on the canvas: V, with a pin on top labelled with the basis vector e-alpha, and P, with a hole in its bottom edge labelled with the basis covector e-alpha, a circled times sign between them. — A vector V^α and a covector P_β. Nothing is connected: side by side already means outer product, and the canvas marks it with ⊗.

Every tensor is one block, and the block is the whole recipe of it:

the body carries the components — the symbol and its indexes, V^α, g_μν;
an upper index (contravariant) is a pin standing on top of the block, drawn in violet, and the pin is labelled e⃗_α;
a lower index (covariant) is a hole in the bottom edge, labelled ẽ^α;
a block with no connector at all is a scalar — a smooth brick, a number.

Those labels are the point, not decoration. The connectors are the basis: the pin of a vector is the basis vector e⃗_α and the hole of a covector is the basis covector ẽ^α. Body and connectors together are the expansion on that basis — V⃗ = V^α e⃗_α — with the components as the ingredients and the basis as what they are ingredients of (§1.4, §1.7).

Which is why the label's variance is the opposite of the connector's: the component V^α goes with e⃗_α, and P_α with ẽ^α. It is not a sign slip; it is the pairing.

A connection is always a pin into a hole: one index up with one index down — a basis vector meeting its dual. That is a contraction, and a contraction is a sum over the repeated index, not a cancellation of symbols. When two blocks fit, the connectors disappear and the bodies merge, multiplying each other (§1.7): the basis is spent in the pairing, and what is left is components. Two pins, or two holes, simply do not go together, and Tessera refuses the fit rather than drawing something false.

Widths are quantised: a block is exactly as many columns wide as it has indexes. The ruler along the top counts those columns, which is why two blocks fitted under a third never overlap.

EQUATION 1 Lowering an index

The metric lowers an index: V_α = g_αμ V^μ. Two clicks and one drag.

In the palette on the left, under fundamental, click g. Then, under generic, click V. Both blocks land in the active term — the one inside the violet band.

g_αβ has two holes, ẽ^α and ẽ^β — both its indexes are down — and V^γ has one pin, e⃗_γ. Tessera picks the name γ on its own: an index that repeated by accident, without being connected, would not be a valid term.
Drag V until its pin meets the right-hand hole of g. As you drag, the target columns light up — green if the core accepts the fit, red if it refuses — and when they are close enough the block snaps into place. When a fit is refused, the panel on the right says why, with the reference to the book.

The pin is in, and the two labels are gone with it — ẽ^γ met e⃗_γ and the pair was spent. What is left is in the bodies: γ appears twice, once up and once down, so it is a dummy index standing for a sum over γ = 1…n. Only ẽ^α is still free, so the assembly is a covector — the index has been lowered.

The panel on the right reads the assembly back to you the whole time: valence, rank, which indexes are free and which are dummy, and what can be done next.

EQUATION 2 Writing it instead of building it

Under by text, type the expression and press Build:

V_{\alpha} = g_{\alpha\mu} V^{\mu}

The whole equation on the canvas: the covector V on the left side, an equals sign, and on the right the metric g with the vector V fitted underneath. — The same expression as before, now a whole equation. The arrangement is automatic: left side, =, right side, all on one line, in the order you read it.

Two things are worth noticing.

Connections need not be stated. An index that appears once up and once down is dummy, and dummy means contracted — so g_{\alpha\mu} V^{\mu} is born already fitted together, with no dragging at all.
The name of a dummy index is arbitrary. It was γ when you built it by hand and μ when you typed it: same expression, different spelling.

Create block adds a single block to the active term; Build reads the whole expression and replaces the line. Both accept LaTeX or the glyph directly — g_{\mu\nu} and g_{μν} give the same block.

Something larger, with two terms and a coefficient — Einstein's tensor:

G_{\alpha\beta} = R_{\alpha\beta} - \tfrac{1}{2} g_{\alpha\beta} R

Einstein's tensor on the canvas: G on the left side, then an equals sign, the Ricci tensor as the first term, and minus one half times the metric times the Ricci scalar as the second. — A sum has no block of its own — it is arrangement. Each summand is a term, laid out side by side with the sign and the coefficient between them. Within a term, blocks placed next to each other are already multiplied: that is the outer product.

EQUATION 3 Letting Tessera do the algebra

Build the metric against its own inverse:

g_{\alpha\mu} g^{\mu\beta}

The metric g with lower indexes alpha and mu, with the inverse metric g with upper indexes mu and beta fitted into its right-hand hole. — The pin μ of g^μβ is already in the hole μ of g_αμ, because μ appears once up and once down.

Now press Simplify in the panel on the right.

A single block remains: delta with a lower index alpha and an upper index beta, one hole and one pin. — Two blocks collapse into one, with the free indexes they had: a hole for α and a pin for β.

And the panel says why it was allowed to do that:

g⁻¹ and g are inverse: the inner product gives back the identity I, whose components are δ. Note that I is a THIRD tensor, neither g⁻¹ nor g. §2.13, eq.2.28

Every step of a simplification comes with its justification and its place in the book. The program does not ask to be believed.

EQUATION 4 The scalar product, and what a sum means

The book writes a covector as components on a basis, P̃ = P_α ẽ^α, a vector as V⃗ = V^α e⃗_α, and their scalar product as P_α V^α. On the canvas both halves of that sentence are already drawn: the components in the bodies, the basis on the connectors. So the product is one gesture — the pin of V into the hole of P, which is ẽ^α meeting e⃗_α.

P_{\alpha} V^{\alpha}

The covector P with the vector V fitted into its hole from below, no connector left free. — Nothing is left free: no pin, no hole. The assembly is a smooth block — a number.

And this is what the panel says the moment the two indexes meet:

index α now appears twice: that triggers a SUM over α = 1…n, not a cancellation of symbols.

That sentence is the book's second picture — the interlocking cubes, P₁V¹ + P₂V² + … + P_nVⁿ. Tessera draws the structure and never the components: the dimension n box at the top of the screen is the n of that 1…n, and the sum over components is what a connection means, not something the mosaic spells out. A sum has no block of its own (§2.3) — and a sum over components has no drawing at all.

Why does the product come out as that plain sum, with the basis nowhere in it? Because the two bases are dual: ⟨ẽ^β, e⃗_α⟩ = δ^β_α — one when the indexes match, zero otherwise. That is what the connectors were doing when they vanished, and you can watch it on its own: the palette has the basis blocks by themselves. Under bases, click ẽ and e⃗, then fit the pin into the hole.

Two basis blocks fitted one under the other: delta with upper beta and lower nu, over delta with upper nu and lower alpha. — In the book the body of a basis block looks empty, but §1.7 warns that it really holds 0…1…0 — that is, δ^·_κ. So Tessera keeps the basis label apart from the connector index, and the fitted pair reads δ^β_ν δ^ν_α: literally the three-step figure on p.18.

Press Simplify.

A single smooth block with no connectors: delta with upper beta and lower alpha. — A smooth block, with no connectors at all — the scalar promised back in section one.

⟨ẽ^β, e⃗_α⟩ is the duality condition: the result is the smooth block δ^β_α, a number. §1.7, eq.1.8

One basis block against a tensor: a single component

Pair a basis block with a covector instead of with its dual, and you get P̃(e⃗_α) = P_α — the α-th component. Drop P, drop e⃗, fit the pin into the hole.

The covector P with a basis-vector block fitted into its hole from below; the basis block's body reads delta with upper lambda and lower alpha. — The basis block wears its body openly: δ^λ_α, the 0…1…0 of §1.7. Both connectors are spent in the fit, so nothing is left free.

Press Simplify.

A single smooth block reading P with a lower alpha, with no connectors at all. — One smooth block: P_α. The α that survives sits in the body, where the basis block kept its own label — it is the name of a compartment, not a connector, and that is why the block is smooth.

P applied to e⃗_α gives back ONE component: P_α, a number. Both connectors were spent in the fit and none is left — the index that remains is the name of a compartment, in the body, not a connector. §1.7, eq.1.4, eq.1.10

The same thing from the other side gives the components of a vector: V⃗(ẽ^α) = V^α (eq.1.10). Rank 2 and up is left alone: there the result is a slice with one index held fixed, which the book does not introduce here, and Tessera does not claim what the book has not said.

MORE Four more worth building

Same three moves — drop, fit, press a button. These are only bigger.

The covariant derivative, with Γ in plain sight

Build \nabla_{\mu} V^{\nu} and press Expand ∇.

∇_μV^ν = ∂_μV^ν + Γ^ν_μλV^λ (§4.14, eq.4.33): the partial plus one Christoffel term per index — +Γ for an upper index, −Γ for a lower one. Both are drawn dashed, because neither ∂ nor Γ is a tensor. The book leaves Γ implicit; here it shows, because hiding it is the habit the program exists to fight.

Ricci, as a contraction of Riemann

Open ready-made assemblies in the palette and pick the Ricci tensor.

The Ricci tensor on the left side, and on the right the Riemann tensor with its own pin plugged into its own third hole, marked by a dashed arc. — R_βν = R^μ_βμν (§5.13, eq.5.52). The dashed arc is a block contracted with *itself* — dragging cannot do that, because a block is rigid, so internal contractions come as a list when you click the block. In Riemann each option says whether it gives Ricci, −R or zero.

A condition: parallel transport

The gradient with the tangent vector U fitted into it, times V, equals a null block with one pin. — ∇_UV⃗ = 0 (§5.4, eq.5.18). Set to zero creates on the other side the null tensor with the *same* free connectors — without that the index balance would not close. Transport U⃗ itself instead of V⃗ and the same assembly is a geodesic.

Changing basis: one Λ per index

Click a block and choose change basis.

The tensor T with an upper and a lower index, with a basis converter fitted above and another below, both drawn with dashed outlines and primed indexes. — §3.1, eq.3.10 asks for as many Λ as upper indexes and as many as lower, so Tessera shoes one onto every free connector. The prime marks the new basis, and the dashes say Λ is not a tensor — the fit is a shoe, not an inner product. Shoe it again and the shoe comes off: the two matrices are inverse to each other.

Then press Simplify, and the shoes are absorbed:

A single block reading T with a primed upper alpha and a primed lower mu, its pin labelled with the primed basis vector and its hole with the primed basis covector. — Λ^α'_α T^α_μ … = T^α'_μ' (eq.3.10). One block again, and the connectors say which basis you are in. Mind what the panel insists on: it is the *same* tensor — what changed are the components, not the object. Λ is not a tensor, and this was a change of description.

NEXT Where to go from here

ready-made assemblies — a list in the palette with twelve equations from the book, built block by block: from g g⁻¹ = δ to the field equation, by way of Bianchi, Ricci, parallel transport and the geodesic.
Click a block and the right panel offers what can be done to it alone: its own internal contractions, a change of basis, the split into symmetric and skew-symmetric parts, its inverse. Shift+click a second one and it enumerates every distinct way the two fit together — the results differ, so the choice is yours to make (§2.7).
Delete removes the selected block, Ctrl+Z undoes the whole assembly, and Clear canvas starts over. A drag counts as one step.
Save writes the assembly to a .json that comes back here, LaTeX copies the equation to paste into a text, and SVG downloads the drawing. Your session is kept in the browser and survives a reload.
Languages — the picker at the top switches between Portuguese, English, Italian and French. The English follows Bernacchi's own vocabulary, so the words on screen are the words in the book.

Every figure on this page was drawn by Tessera itself, by the same code that draws the canvas in your browser — they are the assemblies, not screenshots of them.