an encyclopedia of finite element definitions

# Degree 1 dual polynomial on a dual polygon

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In this example:
• $$R$$ is the reference dual polygon. The following numbering of the subentities of the reference is used:
• Basis functions:
$$\displaystyle \phi_{0} = \begin{cases} \tfrac{4 x}{5} - \tfrac{2 \sqrt{10} y}{5 \sqrt{\sqrt{5} + 5}} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{4 x}{5} - \tfrac{3 y \sqrt{10 \sqrt{5} + 50}}{10} - \tfrac{y \sqrt{2 \sqrt{5} + 10}}{2} + \tfrac{\sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{5} + \tfrac{3 \sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{25} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\tfrac{- 4 x + \tfrac{12 \sqrt{5} x}{5} - \tfrac{2 y \sqrt{10 \sqrt{5} + 50}}{5} - \tfrac{3 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{10 - 2 \sqrt{5}} + \tfrac{8 \sqrt{5}}{5}}{-10 - \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \tfrac{4 x \sqrt{5 - \sqrt{5}}}{5} + \tfrac{4 x \sqrt{\sqrt{5} + 5}}{5} - \tfrac{4 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\- \tfrac{x}{5} + \tfrac{\sqrt{5} x}{5} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\- \tfrac{x}{5} + \tfrac{\sqrt{5} x}{5} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\tfrac{\tfrac{8 \sqrt{5} x}{5} + \tfrac{2 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{2 \sqrt{5} + 10} + \tfrac{3 y \sqrt{10 \sqrt{5} + 50}}{5} + 4 + \tfrac{12 \sqrt{5}}{5}}{\sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\tfrac{- \tfrac{4 x \sqrt{5 - \sqrt{5}}}{5} + \tfrac{4 x \sqrt{\sqrt{5} + 5}}{5} + \tfrac{4 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\\tfrac{4 x}{5} - \tfrac{y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{y \sqrt{10 \sqrt{5} + 50}}{10} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{4 x}{5} + \tfrac{2 \sqrt{10} y}{5 \sqrt{\sqrt{5} + 5}} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{1} = \begin{cases} \tfrac{\sqrt{10} y + 15 \sqrt{2} y + 2 \cdot \left(1 - x\right) \sqrt{\sqrt{5} + 5}}{10 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\- \tfrac{x}{5} - \tfrac{\sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{2} - \tfrac{9 \sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{50} + \tfrac{9 y \sqrt{10 \sqrt{5} + 50}}{20} + \tfrac{5 y \sqrt{2 \sqrt{5} + 10}}{4} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\tfrac{- \tfrac{8 \sqrt{5} x}{5} + 16 x + \tfrac{2 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{2 \sqrt{5} + 10} + \tfrac{3 y \sqrt{10 \sqrt{5} + 50}}{5} + \tfrac{8 \sqrt{5}}{5}}{-10 - \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{\tfrac{4 x \sqrt{\sqrt{5} + 5}}{5} + \tfrac{16 x \sqrt{5 - \sqrt{5}}}{5} + 2 \sqrt{2} y + \tfrac{6 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\- \tfrac{x}{5} + \tfrac{\sqrt{5} x}{5} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\- \tfrac{x}{5} + \tfrac{\sqrt{5} x}{5} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\tfrac{\tfrac{8 \sqrt{5} x}{5} + \tfrac{2 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{2 \sqrt{5} + 10} + \tfrac{3 y \sqrt{10 \sqrt{5} + 50}}{5} + 4 + \tfrac{12 \sqrt{5}}{5}}{\sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\tfrac{- \tfrac{4 x \sqrt{5 - \sqrt{5}}}{5} + \tfrac{4 x \sqrt{\sqrt{5} + 5}}{5} + \tfrac{4 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\- \tfrac{x}{5} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{20} + \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{20} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \sqrt{10} y + 5 \sqrt{2} y + 2 \cdot \left(1 - x\right) \sqrt{\sqrt{5} + 5}}{10 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{2} = \begin{cases} \tfrac{- 5 \sqrt{2} y + \sqrt{10} y + 2 \cdot \left(1 - x\right) \sqrt{\sqrt{5} + 5}}{10 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\- \tfrac{x}{5} - \tfrac{y \sqrt{2 \sqrt{5} + 10}}{4} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{20} + \tfrac{\sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{50} + \tfrac{\sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\tfrac{- \tfrac{28 \sqrt{5} x}{5} - 4 x - 3 y \sqrt{10 - 2 \sqrt{5}} - y \sqrt{2 \sqrt{5} + 10} + \tfrac{3 y \sqrt{10 \sqrt{5} + 50}}{5} + \tfrac{7 y \sqrt{50 - 10 \sqrt{5}}}{5} + \tfrac{8 \sqrt{5}}{5}}{-10 - \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \tfrac{16 x \sqrt{\sqrt{5} + 5}}{5} - \tfrac{4 x \sqrt{5 - \sqrt{5}}}{5} - 2 \sqrt{2} y + \tfrac{6 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\- \tfrac{6 x}{5 + 5 \sqrt{5}} + \tfrac{\sqrt{2} y}{\sqrt{5 - \sqrt{5}}} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\\tfrac{- 6 x \sqrt{5 - \sqrt{5}} + 5 \sqrt{2} y \left(1 + \sqrt{5}\right) + \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\tfrac{\tfrac{8 \sqrt{5} x}{5} + \tfrac{2 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{2 \sqrt{5} + 10} + \tfrac{3 y \sqrt{10 \sqrt{5} + 50}}{5} + 4 + \tfrac{12 \sqrt{5}}{5}}{\sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\tfrac{- \tfrac{4 x \sqrt{5 - \sqrt{5}}}{5} + \tfrac{4 x \sqrt{\sqrt{5} + 5}}{5} + \tfrac{4 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\- \tfrac{x}{5} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{20} + \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{20} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \sqrt{10} y + 5 \sqrt{2} y + 2 \cdot \left(1 - x\right) \sqrt{\sqrt{5} + 5}}{10 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{3} = \begin{cases} \tfrac{- 5 \sqrt{2} y + \sqrt{10} y + 2 \cdot \left(1 - x\right) \sqrt{\sqrt{5} + 5}}{10 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\- \tfrac{x}{5} - \tfrac{y \sqrt{2 \sqrt{5} + 10}}{4} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{20} + \tfrac{\sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{50} + \tfrac{\sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\tfrac{- 4 x + \tfrac{12 \sqrt{5} x}{5} - \tfrac{2 y \sqrt{10 \sqrt{5} + 50}}{5} - \tfrac{3 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{10 - 2 \sqrt{5}} + \tfrac{8 \sqrt{5}}{5}}{-10 - \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \tfrac{4 x \sqrt{5 - \sqrt{5}}}{5} + \tfrac{4 x \sqrt{\sqrt{5} + 5}}{5} - \tfrac{4 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\- \tfrac{6 x}{5 + 5 \sqrt{5}} - \tfrac{\sqrt{2} y}{\sqrt{5 - \sqrt{5}}} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\\tfrac{- 6 x \sqrt{5 - \sqrt{5}} - 5 \sqrt{2} y \left(1 + \sqrt{5}\right) + \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\tfrac{- \tfrac{52 \sqrt{5} x}{5} - 20 x - \tfrac{2 y \sqrt{10 \sqrt{5} + 50}}{5} - \tfrac{3 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{10 - 2 \sqrt{5}} + 4 + \tfrac{12 \sqrt{5}}{5}}{\sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\tfrac{- \tfrac{16 x \sqrt{\sqrt{5} + 5}}{5} - \tfrac{4 x \sqrt{5 - \sqrt{5}}}{5} - \tfrac{6 \sqrt{10} y}{5} + 2 \sqrt{2} y - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\- \tfrac{x}{5} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{20} + \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{20} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \sqrt{10} y + 5 \sqrt{2} y + 2 \cdot \left(1 - x\right) \sqrt{\sqrt{5} + 5}}{10 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{4} = \begin{cases} \tfrac{- 5 \sqrt{2} y + \sqrt{10} y + 2 \cdot \left(1 - x\right) \sqrt{\sqrt{5} + 5}}{10 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\- \tfrac{x}{5} - \tfrac{y \sqrt{2 \sqrt{5} + 10}}{4} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{20} + \tfrac{\sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{50} + \tfrac{\sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\tfrac{- 4 x + \tfrac{12 \sqrt{5} x}{5} - \tfrac{2 y \sqrt{10 \sqrt{5} + 50}}{5} - \tfrac{3 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{10 - 2 \sqrt{5}} + \tfrac{8 \sqrt{5}}{5}}{-10 - \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \tfrac{4 x \sqrt{5 - \sqrt{5}}}{5} + \tfrac{4 x \sqrt{\sqrt{5} + 5}}{5} - \tfrac{4 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\- \tfrac{x}{5} + \tfrac{\sqrt{5} x}{5} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\- \tfrac{x}{5} + \tfrac{\sqrt{5} x}{5} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\tfrac{\tfrac{28 \sqrt{5} x}{5} + 20 x - \tfrac{7 y \sqrt{10 \sqrt{5} + 50}}{5} - 3 y \sqrt{2 \sqrt{5} + 10} - \tfrac{3 y \sqrt{50 - 10 \sqrt{5}}}{5} - y \sqrt{10 - 2 \sqrt{5}} + 4 + \tfrac{12 \sqrt{5}}{5}}{\sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\tfrac{\tfrac{4 x \sqrt{\sqrt{5} + 5}}{5} + \tfrac{16 x \sqrt{5 - \sqrt{5}}}{5} - \tfrac{6 \sqrt{10} y}{5} - 2 \sqrt{2} y - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\- \tfrac{x}{5} - \tfrac{7 y \sqrt{2 \sqrt{5} + 10}}{20} + \tfrac{y \sqrt{10 \sqrt{5} + 50}}{20} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- 15 \sqrt{2} y - \sqrt{10} y + 2 \cdot \left(1 - x\right) \sqrt{\sqrt{5} + 5}}{10 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$