How can I draw a water lily in LaTeX?

You can use \clip to cut away the outside parts, and use polar coordinates.

\documentclass[tikz,border=3mm]{standalone}
\begin{document}
\begin{tikzpicture}[declare function={R=3;alpha=-20;},thick]
 \draw circle[radius=R]; 
 \clip circle[radius=R]; 
 \draw foreach \X in {0,...,6}
 {(alpha+60*\X:R) circle[radius=R]};
\end{tikzpicture}
\end{document}

Are you looking for this 5-petal "water-lily" ?

% a 5-petal rose (or "water-lily" if you like ^^)
\documentclass[tikz,border=3mm]{standalone}
\begin{document}
\begin{tikzpicture}
\def\R{3}   
\draw (0,0) circle(\R); 
\draw[smooth,magenta] plot[domain=0:36*5,samples=200] (\x:{\R*cos(5*\x)});
\end{tikzpicture}
\end{document}

I update the Asymptote version of @Jairo

//http://asymptote.ualberta.ca/
unitsize(3cm);
draw(unitcircle);
path petal=(1,0) .. (0,0) .. dir(144);
for(int i=1; i<=5; ++i) {draw(rotate(72*i+30)*petal,red);}

Using LuaTeX, it is easy to generate some more generic results.

\documentclass{article}
\usepackage{tikz}
\usepackage{luacode}

\begin{document}

\tikzset{
  arcstyle/.style={
    thick
  }
}

\begin{luacode*}
one_degree = math.pi / 180

function get_inscribed_point(radius, n_poly, index, rotation)
  local ang = (360.0 / n_poly * index + rotation) * one_degree
  local x = radius * math.cos(ang)
  local y = radius * math.sin(ang)
  return {x,y}
end


function get_arc_info(p1, p2, p3)
  local xa, ya = table.unpack(p1)
  local xb, yb = table.unpack(p2)
  local xc, yc = table.unpack(p3)
  
  local coef1 = xb*xb - xc*xc + yb*yb - yc*yc
  local coef2 = xa*xa - xb*xb + ya*ya - yb*yb
  local coef3 = 2.0 * ((xa-xb)*(yb-yc)-(xb-xc)*(ya-yb))
  
  -- calculate center
  local center_x = (-(ya-yb)*coef1+(yb-yc)*coef2)/coef3
  local center_y = ((xa-xb)*coef1-(xb-xc)*coef2)/coef3
  
  -- calculate radius
  local radius = math.sqrt(math.pow(xa-center_x, 2)+math.pow(ya-center_y, 2))
  
  -- calculate arc angle range
  local arc_ang1 = math.asin((0.5*math.sqrt(math.pow(xb-xa,2)+math.pow(yb-ya,2))/radius))
  local arc_ang2 = math.asin((0.5*math.sqrt(math.pow(xc-xb,2)+math.pow(yc-yb,2))/radius))
  local arc_angle = (arc_ang1 + arc_ang2) * 2.0
  
  -- find out if (xc,yc) or (xa,ya) has the smallest angle
  -- make sure xc has the smallest angle (if not, swap two points)
  local ang_a = math.atan2(ya-center_y,xa-center_x)
  local ang_c = math.atan2(yc-center_y,xc-center_x)
  if ang_a < ang_c then
    ang_a = ang_a + 2 * math.pi
  end
  
  -- determine start angle
  local start_angle =ang_c
  local end_angle = ang_a
  
  -- return results
  return {
    ["center_x"] = center_x,
    ["center_y"] = center_y,
    ["radius"] = radius,
    ["start_angle"] = start_angle / one_degree,
    ["end_angle"] = end_angle / one_degree,
    ["arc_start_x"] = xc,
    ["arc_start_y"] = yc
  }
end

function draw_arc(p1, p2, p3)
    local arc = get_arc_info(p1, p2, p3)
      
    tex.print(string.format([[\draw[arcstyle] (%f cm, %f cm) arc (%f:%f:%f cm);]],
      arc["arc_start_x"],
      arc["arc_start_y"],
      arc["start_angle"],
      arc["end_angle"],
      arc["radius"]))
end

function draw_lily(radius, n_poly, offset, rotation)
  for i=1,n_poly do
    local ind1 = i - 1
    local ind2 = (i+offset - 1)%n_poly
    local p1 = get_inscribed_point(radius, n_poly, ind1, rotation)
    local p2 = {0.0,0.0}
    local p3 = get_inscribed_point(radius, n_poly, ind2, rotation)
    draw_arc(p1, p2, p3)
  end
end

\end{luacode*}


\begin{tikzpicture}
\draw (0,0) circle (2cm);
\directlua{
  draw_lily(2.0, 6, 2, 0.0)
}
\end{tikzpicture}
\begin{tikzpicture}
\draw (0,0) circle (2cm);
\directlua{
  draw_lily(2.0, 6, 2, 15.0)
}
\end{tikzpicture}
\begin{tikzpicture}
\draw (0,0) circle (2cm);
\directlua{
  draw_lily(2.0, 8, 2, 15.0)
}
\end{tikzpicture}
\begin{tikzpicture}
\draw (0,0) circle (2cm);
\directlua{
  draw_lily(2.0, 12, 2, 15.0)
}
\end{tikzpicture}
\begin{tikzpicture}
\draw (0,0) circle (2cm);
\directlua{
  draw_lily(2.0, 12, 3, 15.0)
}
\end{tikzpicture}
\begin{tikzpicture}
\draw (0,0) circle (2cm);
\directlua{
  draw_lily(2.0, 12, 4, 15.0)
}
\end{tikzpicture}
\begin{tikzpicture}
\draw (0,0) circle (2cm);
\directlua{
  draw_lily(2.0, 36, 2, 15.0)
}
\end{tikzpicture}

\end{document}