Hearty Pattern!

There are $14$ large sectors and $14$ small sectors, where the large ones overlap. There are several cases to consider: the heart is formed by either overlapping sectors or tangential sectors.

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Tangential Sectors

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For small sectors , the approach is trivial. Since there are clearly $14$ pairs of adjacent sectors that each form a small heart, we count $14$ possibilities.

There are two regular heptagons in one regular $14$ -gon.

For large sectors , since the large circles overlap two segments of a regular $14$ -gon, known as tetradecagon , then counting the number of hearts formed by two tangential large circles is equivalent to counting the number of sides of $7$ -gon or heptagon . In this case, there are $7$ possibilities. Thus, since there are $2$ regular heptagons enclosed by a regular $14$ -gon, there are $7 \cdot 2 = 14$ possibilities.

So we counted $28$ hearts of tangential sectors.

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Overlapping Sectors

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For different-sized sectors , consider the following diagram:

Three sectors indicated by the bolded lines

Because there are $14$ large sectors adjacent to $2$ small sectors, we counted $28$ more hearts.

For two overlapping large sectors , let's now look at the following diagram:

Two large sectors indicated by the bolded lines

The common region shared by the two sectors indicate the number of hearts, which is congruent to counting the number of midpoints of each segment of $14$ -gons. Because there are $14$ midpoints, then there are $14$ more hearts.

Overall, we counted $42$ hearts.

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Answer

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Combining the numbers of hearts from both previous sections, we have $28 + 42 = \boxed{70}$ hearts.